A Comprehensive Analysis Structure for the Design of Masonry Arches

1.1.1. Removing Sudden Collapse

The removal of the sudden collapse has been proposed through Reinforced Stability-Based Design (RSBD) [66]. This RSBD technique maintains the traditional SBD behavior under normal service conditions and designs a reinforcement system that “activates” after the loss of stability. Failure is quantified by defining it as the loss of stability, and safety is added through the application of secondary reinforcement. The first mathematical investigation into this behavior produced a linear relationship between the applied load and mechanical hinge rotation for small angles. This linearity is carried into a linear strength problem through the application of a linearly elastic tensile material. The proposed methodology is achieved through the strategic application of reinforcement to remove the no-slip assumption, define the failure mechanism, and its resistance to motion.

The RSBD approach creates an efficient and accessible methodology to removing the conditions of slippage and sudden collapse. The application, however, only addresses the kinematically driven failure. What’s missing is an analysis structure that efficiently and effectively quantifies the system through the various stages of development and design.

1.3.1. Kinematic Equilibrium

The static equilibrium analysis of a kinematic state superimposed on a stable condition through the incorporation of at least one external loading variable to the free-body diagram and corresponding equations of equilibrium that define the structure.

The novelty of this work is the establishment of a comprehensive analysis structure for masonry arches that is designed for the efficient and accessible control, management, and quantification of a defined failure condition. This work begins with the establishment of kinematic equilibrium, its application to masonry arches subjected to mechanical joint control, and its subsequent use in all the developed static and dynamic analysis procedures, methodologies, and software.

2.1.1. Equilibrium Problem

Statics states that a structure or its members are in equilibrium when the forces and moments are balanced. For principal load-carrying portions that lie on a single plane, the Equations of Equilibrium (EOE) are reduced to where ∑F_x and ∑F_y are the algebraic sums of the cartesian components of the forces and ∑M_O is the sum of the moments about the point O.

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For the traditional rigid-no-tension model of a masonry arch, the standard kinematically admissible mechanism requires the development of four hinges that alternate between the intrados and extrados (Fig. 2b). A hinge develops when the concentration of the normal forces near the intrados or extrados and can be idealized by concentrated point loads at the boundary. Assuming perfect hinges allows the kinematic arch to be expressed as three rigid pin connected elements, as shown in Fig. (4).

A pin connection between blocks removes the ability to carry a moment at the joint. Decomposing (Fig. 4) into the three rigid elements as seen in Fig. (5) allows the standard static equilibrium equations (i.e., Eq. (1)) to be constructed for the three-element systems. Note in Fig. (5) that the decomposition of the three rigid elements includes the lever arm distances used for calculating moments to satisfy Eq. (1).

The three-element representation of the arch generates nine equilibrium equations, but under its self-weight, only eight unknowns exist due to the pin connections. In the traditional sense of statics, this condition is unstable. A ninth variable is required to create a determinate system and with it, a single solution to the equilibrium problem. This ninth variable can either be a restriction of motion through the removal of a pin connection (i.e., the three-pinned arch) or the inclusion of an applied force that imposes equilibrium on the system.

It is important to note that a kinematic system at rest and in equilibrium is static. While this notion is unfavorable in the context of civil engineering, the unidirectional motion of the defined kinematic arch allows the condition to be specific to a single condition that can be controlled through design.

Assuming stability under self-weight, therefore, requires the application of an external load to generate a kinematic condition. Thus, the loading condition can be assigned a variable and incorporated into the EOE to construct a kinematically determinant system with a single solution.

In matrix form, the EOE can be expressed as

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where BC is the balanced matrix, r is the reaction vector, and q is the constants vector. The addition of a loading condition and the accompanying loading variable into the formation of the EOE generates a non-zero determinant for BC, allowing the reaction variables to be obtained by

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The EOE can thus be represented by a combination of a balanced matrix, reaction vector, and constants vector group required to solve Eq. (3).

2.1.2. Loading Conditions

A loading condition variable is introduced to the EOE by adding a multiplier to the geometry of the applied condition. The choice of the multiplier depends on the conditions under evaluation. For seismic assessments, the multiplier, λ_a, is a factor of the gravitational constant g, which simplifies the body forces. Fig. (6) shows the free body diagram for a uniform acceleration applied to the four-pinned arch in Fig. (4). The uniform acceleration generates the EOE

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In Eq. (4) v_i and h_i are the vertical and horizontal reactions at the i^th hinge, respectively, and f_gj is the body force of the j^th element applied at the element’s center of mass. The moments were calculated by the cartesian forces multiplied by their respective vertical, Δy, and horizontal Δx, lever arms. The subscripts of, Δx, and Δy, denote the hinges or center of mass locations used to construct the lever arms and as shown in Fig. (5) (i.e., Δy_2,1 is (y₂ – y₁) and Δx_1,CM1 is (x₁ – x_CM1)). The acceleration vector is

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where λ_a is the magnitude and θ_a is the polar angle of the acceleration. Note that either a direction or magnitude of the acceleration is required to maintain the linear set of equations. If the angle is defined, then the variable is λ_a. If λ_a is fixed, the evaluation can determine rotational collapse. This rotational collapse is useful for simplified seismic evaluations and experimentation (see Section 2.3.2.2).

2.1.2.1. Combining loads

Eqs. (4 and 5) represent the equilibrium condition shown in Fig. (4). Different loading conditions require different EOE sets, but within the structure of BC, the effects of different loading variables are isolated in a single column, as highlighted in Fig. (7). This allows for a simple exchange between q and the load variable column of BC. This is important in the context of new construction as it identifies a simple methodology that can be utilized to establish a simple database structure for the required loading conditions that must be analyzed for limiting conditions and structural detailing.

2.2.1. Kinematic Admissibility

The admissibility of a given loading-mechanism condition is subdivided into a two-stage evaluation. The first is a direct assessment of r. The four-hinged mechanism under uniform acceleration requires a positive λ_a and compressive forces at the hinges. The compressive force evaluation is achieved by comparing the net reaction force vector direction at each hinge with the boundary line of the mechanical joint.

The second stage of the analysis process is an evaluation of the thrust line geometry. The thrust line is a theoretical line that represents the flow of compressive forces. Stability requires this line to exist within the material geometry. For the admissibility of a defined kinematic condition, it only has to pass through the defined hinges.

2.2.2. Kinematic Collapse Load Calculator

Both the kinematic EOE and the thrust line are geometrically dependent. The solution to Eq. (3) also requires the inverse matrix calculation of BC. These conditions are too labor-intensive to perform by hand, but modern computing power and drafting software make these conditions almost trivial and immediate. Thus, the creation of the KCLC [70].

The KCLC is a stand-alone interactive calculator developed in MATLAB^® to perform the black-box analysis of masonry arches. The original open-source KCLC utilizes the kinematic EOE and kinematic admissibility conditions to provide an analysis of user-defined circular arches subjected to either an asymmetric point load or constant horizontal acceleration. It allows the adjustment of the defined kinematic mechanism through hinge positioning; displays the arch, hinges, and loading condition; and calculates and displays the solution to r and the thrust line (Fig. 10). For each change in the hinge position, r and the thrust line are recalculated and displayed.

The original KCLC requires no direct understanding of the analysis, which enables it to become an effective educational tool for teaching the concepts of the kinematic theorem, the thrust line, and stability. Adaptations to the original software have also significantly broadened the conditions that can be analyzed. These expansions include additional mechanisms, generic arch geometries, localized capacity compensation requirements to establish traditionally non-stable but kinematically admissible conditions, and mechanical deformations.

2.2.2.1. KCLC and Multi-Mechanism Analysis

The first adaptations to the KCLC were implemented to address an experimentally observed non-traditional mechanism [71]. The observed mechanism replaced hinge H₁ with a slip joint. Six additional mechanism types were incorporated into the software. The added mechanism types included the different combinations of slip joints, hinge joints, and combined slip-hinge joints, paired with the observed slip at H₁. From the inclusion of the observed failure condition into the KCLC software: (1) a friction angle consistent with the block material was calculated for measured capacity of the arch; and (2) the observed mechanism was the limiting condition of the seven different mechanisms [69].

These added mechanism types are only a fraction of the possibilities. In total, there exists the potential of 70 different mechanism types [68]. It will be necessary to evaluate all the possible failure modes when designing an arch for civil applications, but the database structure of the EOE combined with the KCLC creates an incredibly efficient analysis, as shown with the inclusion of the seven [69].

2.2.3. Capacity Compensation for Non-Stable Mechanisms

Kinematic admissibility only requires the boundary conditions of motion (i.e., hinge locations, mass distribution, and loading geometry). This allows the theoretical thrust line to exist outside the material between hinges. The laws governing forces disagree. For any kinematic condition to physically exist, the thrust line has to be adjusted back within the material boundary to maintain the assumption of rigid elements between mechanical joints. Theoretically, this is achieved by imposing an eccentric shift at each block joint where the thrust line lies outside the material boundary and balancing it with a required moment or opposing tensile force at the specified position. The thrust point positions calculated through the free-body diagrams for thrust point calculations (Section 2.2.1.1) establish a quantifiable condition for this eccentric shift. A more detailed description can be found in the literature [68, 69]. The importance here is that the minimum reinforcing capacity requirements necessary to define a particular mechanism can be efficiently quantified and incorporated into the KCLC. This will ultimately support the efficient detailing of the engineered system.

2.2.4. Mechanical Deformations

The final consideration of kinematic equilibrium is the actual mechanical deformations from the defined kinematic condition. An arch can be mechanically deformed and static. This deformation can arise from foundation settlements, finite reinforcement stiffness, and loading conditions that initiate mechanization but do not drive it to collapse. Each of these results in changes to the boundary conditions, but kinematic equilibrium and admissibility still hold. The ability to impose these deformations is therefore necessary.

The kinematic condition of a dry-stack masonry arch is a Single Degree of Freedom (SDOF) problem. The SDOF is confined to the mechanical joints. In the context of design, the mechanism is engineered through reinforcing against non-ideal mechanisms (Section 2.2.2.1) and any capacity compensation requirements (Section 2.2.3).

2.2.4.1. Mechanism Motion

To examine experimentally observed static SDOF deformations, a simplified KCLC was constructed [69, 71]. SDOF motion requires the definition of one degree of motion to express the total deformation of the system. In the context of a dry-stack arch with defined mechanical hinge joints, the arch-hinge configuration is represented by three fixed lengths, l_ij, connected by four pins at the hinge locations (Fig. 11). The SDOF that binds the motion is the horizontal displacements of hinges H₂ and H₃. Hinges H₁ and H₄ are translationally fixed to their respective bases and thus limited to rotations. For a given rotation, Δα₁, at hinge H₁ the rotation of hinge H₄ can be expressed by Eq. (6) through the application of the law of cosines. The lengths, l_ij, and angles, θ_jk, are identified in Fig. (11). The translations of hinges H₂ and H₃ can then be determined through the application of the rotation displacements, Δα₁ and Δα_4, and lengths l₁₂ and l_34, respectively. Lastly, the polar rotation of the H₂-H₃ connection can be determined through the comparison of the initial and final states. This polar change provides the additional rotation to define the intermittent hinges’ deformations.

Applying these mechanical deformations to an arch and reapplying the kinematic EOE in the KCLC structure reveal (1) that static-deformed conditions still require external loading to exist; (2) that the deformation path within a kinematically admissible range of motion can be defined to single-point translation and lever arm rotations; and (3) the evaluation of capacity compensation with the deformations can generate the ability to directly calculate experimentally observed finite hinge stiffness of reinforced joints [74].

This extends the theoretical foundation of the method of masonry to the deformation of the defined condition. The KCLC, though, is only a calculator. It does not provide a strategy or engineering judgement.

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2.3.1. Collapse Load Diagrams

First-order strategies of assessment exist for most modern structural systems. These first-order assessments provide the platform for a project’s planning, development, scope, cost estimates, and allocation of funds. They are a critical component to the successful implementation of any modern structural system. For civil projects, these first-order assessments can often be carried beyond the 60% design stage. For existing systems, first-order assessment strategies streamline inspections and interventions, and their efficiency becomes critical in post-disaster situations.

Collapse Load Diagrams (CLD) are a first-order assessment strategy formulated for masonry by combining a set of capacity results with a single parameter [75]. The single parameter for the arch is the polar angle between the base hinges H₁ and H₄ taken from the intersection of the arche’s base and midspan. The capacity results present the set of minimum collapse loads for all admissible combinations of the base hinges. Plotting these capacities against the negative tangent of the polar angle between the base hinges generates the CLD.

The diagrams are created by fixing the base hinges and determining the minimum positive collapse multiplier for the admissible variations of hinges H₂ and H₃ using the same EOE with updated boundary conditions to account for the change in hinge position. The minimum multiplier and the associated hinge positions are recorded, one base hinge position is changed, and the new minimum multiplier is determined. This is repeated until all admissible base hinge configurations have been exhausted. These values are plotted, and lines are drawn connecting the multipliers associated with one of the two base hinges remaining fixed.

Fig. (12) shows a CLD and arch geometry for a 27-block semi-circular arch subjected to a constant horizontal acceleration. The CLD is accompanied by a subplot of the arch-load geometry, which also identifies the limits of admissible base hinge positions. This provides not only the capacity values but a quick visualization of where the base hinges can be defined in the development of a hinge-controlled arch system. The CLD thus provides the types of capacity values and required base reinforcement criteria ideal for sizing an arch and shaping its mechanism in the preliminary design stages of a project. The KCLC would then be available for the detailing of the selected arch as they both utilize the same foundational EOE.

2.3.2. Tilt Test Characterization

A notable characteristic of the CLDs is that hinge H₁ dominates capacity. The effects of hinge H₄ are secondary to H₁. This unique relationship allows the capacity behavior to be formulated into a single variable approximation (i.e., hinge H₁’s position). This key variable, in combination with experimental capacity measurements of a family of mechanisms, generates a simplified link between a physical arch and its theoretical model. In other words, the arch can be characterized To test this characterization potential, two experimental arches subjected to hinge control were constructed [71, 76]. Both arches were semi-circular with 27 blocks (Fig. 13). The first arch (Fig. 13a) was an in-scale arch and had a clear span of 0.67 m. It was constructed from timber blocks with Velcro^® providing reinforcement for the non-mechanical joints. The mechanical joints were defined by the lack of reinforcement. The second arch (Fig. 13b) was a full-scale arch with a clear span of 3.84 m. The blocks were engineered from the oriented standard board for the block faces and steel risers for the out-of-plane thickness. 36-grit sandpaper was added to the joint faces to increase the friction angle between blocks, and cam straps were the reinforcing material for establishing hinge control.

For each arch, 25 distinct hinge sets were tested with three measured failures per set. The failure capacities of each set were averaged and compared against the theoretical values. Then the hinge set capacities, both theoretical and experimental, were averaged for fixed H₁ positions and then compared again through normalization. From this comparison, a single linear capacity adjustment equation was established for each arch that adjusts the theoretical capacities to the experimental. Fig. (14) shows the capacity adjustment equations for both arches. For the in-scale arch, a DEM analysis was also performed, and a linear capacity adjustment equation was also established (Fig. 14a) [69]. This equation differs from the kinematic equilibrium approach, but a single adjustment equation is established, which highlights the versatility of this characterization approach. For the full-scale arch, there were some observed anchorage issues between the arch and the tilting platform [76]. This was adjusted, and the linear equation was obtained from the final three positions of H₁ (Fig. 14b).

Note that Eq. (5) was restructured such that the rotation angle was the collapse load variable. The actual block measurements were used to draw the model arch in AutoCAD^® and then extracted for the arch analysis model. Also, a CLD was created for each arch to establish the hinge sets tested and theoretical capacities. Refer to the literature for a full description of the experimental campaigns [68, 71, 76]. The key here is that the hinge-controlling of dry-stack arches and the examination of a family of admissible mechanisms allows for the direct characterization of the analysis model to the actual arch. This significance is further entrenched by the lack of damage observed to the arch. This generates the potential to establish individual capacity ratings for constructed arches.