Thrust Line Research Articles

Safety assessment of masonry undamaged and damaged arches subjected to gravitational loads and horizontal forces. A numeric procedure to identify the optimal thrust line

In 2019 the authors formulated a numerical procedure for assessing the safety of masonry arches based on their geometry and the shape of the thrust line. The thrust line is calculated to be nearest to the geometric axis of the arch, because it is that corresponding to the optimal inner stress state. The original procedure, which focused on the safety analysis of undamaged fixed arches subject to gravitational loads, is reformulated here to also consider arches on pinned supports as well as damaged arches, i.e. one- and two-hinged arches. The procedure is further extended to also assess safety against horizontal actions, such as those resulting from an earthquake. Finally, safety is evaluated by identifying the domain of admissible thrust lines within the thickness of the arch and by computing the degree of safety in terms of the capacity of the thrust line to be moved vertically within the domain. The reformulation and extension of the procedure exploits the technique of partitioned matrices to enable automation and easy implementation in structural calculation software.

The thrust line in masonry arches: a new simplified algorithm

Masonry arches and vaults have been widely used in the past to construct horizontal structural elements using materials with low tensile strength. Initially, the design methods used were mostly based on empirical rules derived from the experience of builders. In more recent centuries, however, analytical theories and graphical methods have been developed to formalize the static behaviour of the masonry arch. Although computational methods for evaluating masonry arch structures have been widely developed in recent decades, graphical and analytical methods are still simple and clear tools for understanding the structural behaviour of these structures. Therefore, a simplified method for identifying the thrust line in masonry arches is presented in this paper. The proposed algorithm is then applied to several case studies, and the results obtained are compared with those derived from finite element models.

Numerical-vector succession for the graphic structural analysis of masonry historic buildings with arches and symmetrical systems

Many historic buildings have symmetry in their geometric configuration. The objective of this research is to denote the application of numerical-vector succession in the structural analysis of historical masonry buildings, with arches and symmetrical systems, including mathematical processes in ancient graphic analysis, emphasizing the importance of loads in the structural stability. We based the analysis on three fundamental stages: recognition of the construction system of the heritage object, geometric discretization of the system and vector analysis under different physical considerations. Hence, the thrust lines are affected by the loads, boundary conditions and history of structural behaviour. Numerical and computational tools offer faster and more accurate graphic analysis processes. Many historic buildings have symmetry in their geometric configuration. The objective of this research is to denote the application of numerical-vector succession in the structural analysis of historical masonry buildings, with arches and symmetrical systems, including mathematical processes in ancient graphic analysis, emphasizing the importance of loads in the structural stability. We based the analysis on three fundamental stages: recognition of the construction system of the heritage object, geometric discretization of the system and vector analysis under different physical considerations. Hence, the thrust lines are affected by the loads, boundary conditions and history of structural behaviour. Numerical and computational tools offer faster and more accurate graphic analysis processes.

The Stone Skeleton: A Reappraisal

Jacques Heyman’s 1966 article “The Stone Skeleton” has been influential in the assessment of masonry arches in general, including masonry bridges and gothic architecture. Heyman’s article relies on limiting assumptions about arch behavior, and concludes that an arch can be declared stable based on the location of a statically admissible thrust line within the arch. In the following commentary, we assess the validity of the assumptions made by Heyman and the uses that have been made of his application of the lower bound theorem of plasticity. We conclude that Heyman’s methods have enduring value, but that the user needs to recognize the limitations of the assumptions made and the limits on the validity of conclusions imposed by these assumptions.

Analysis and Optimization Design of U-Shaped Ironless Permanent Magnet Linear Synchronous Motor

The working speed of the U-shaped ironless permanent magnet linear synchronous motor is determined by the frequency and pole pitch. When the motor’s working speed is required to be constant, different combinations of frequency and pole pitch can be chosen. This article mainly discusses the influence of the pole pitch on the motor performance when other parameters are kept constant. First, the magnetic field of the motor permanent magnet is analyzed using the equivalent magnetizing current method to discuss the relationship between the pole pitch and the motor air gap magnetic field. Then, the finite element method is used to analyze the influence of the pole pitch on the motor air gap magnetic field, transverse edge effect, and electromagnetic thrust considering the saturation effect of the magnetic yoke. The research shows that under the conditions of constant motor basic structural parameters and speed, and the same amount of permanent magnet material, both the air gap magnetic density and the motor thrust line density have maximum values as the pole pitch changes. The research results have certain guiding significance for the design of this type of motor.

Experimental analysis of a planing monohull model during forward acceleration: part II – evaluation of an indirect method for applying thrust angle in the experiments

ABSTRACT Most towing tank model tests conducted on planing hulls are not performed under self-propulsion conditions, and it is often not feasible to directly apply towing force at the corresponding point of action and along the thrust of the full-scale vessel. Therefore, it is essential to establish a well-defined method for accurately applying the thrust line in bare hull model tests using alternative loading solutions, such as CG shift, and understand how any deviation affects the model’s running attitude and performance. This article presents an analysis of the dynamic equilibrium equations of motion for the model, considering various variables that can be used to adjust the thrust line in towing tank tests. The proposed formalism can aid in determining the external loading requirements in resistance tests. Finally, the article discusses the challenges and provides an error estimation for a simplified method of applying a constant thrust angle in the accelerated tests.

A novel impact model for the rocking motion of masonry arches

The in-plane rocking motion of a masonry arch subjected to ground acceleration is investigated, focusing on the impacts at stereotomy sections, which may occur during the motion. It is assumed that the arch arrives at the impact moving along a prescribed four-hinge mechanism and that, after the impact, it continues its motion along a new four-hinge mechanism to be determined. The novel concept of impulse line, which is analogous to the thrust line computed during the smooth motion, is introduced to describe the impulsive stress state arising within the arch at the impact. That is the basis for extending the Housner impact model, initially proposed for the rocking motion of a free-standing column, to the more complicated case of a masonry arch behaving as a single-degree-of-freedom system. The mechanism after the impact is determined by minimizing the kinetic energy loss of the arch at impact, i.e. by maximizing its restitution coefficient, over the set of compatible mechanisms that fulfill a suitable formulation of the virtual work principle. The descending impulse line is proven to be equilibrated, kinematically admissible (i.e., not resisting the opening of the hinges after the impact), and statically admissible (i.e., corresponding to a compressive impulsive stress state). Numerical results are presented, discussing the restitution coefficient of discrete and continuous circular arches with parameterized geometry, for which the four-hinge mechanism before the impact is assumed to follow from an equivalent static analysis.

Application of thrust layout optimization to masonry structures

A new automated analysis procedure designed to overcome limitations of the traditional thrust line analysis method for masonry gravity structures is presented. The procedure, termed thrust layout optimization (TLO), is capable of automatically identifying admissible thrust lines in masonry gravity structures comprising general arrangements of masonry blocks. The procedure employs a modified truss layout optimization with transmissible loads formulation, which means that an initial thrust line layout does not need to be specified in advance. Highly visual output is generated, allowing areas where tensile strength has been implicitly assumed to be clearly identified, which is beyond the scope of traditional thrust line analysis. Also, sliding failures can be modelled without difficulty. Finally, examples are used to demonstrate the efficacy of the TLO procedure; these show that thrust line bifurcations can be automatically identified in problems involving openings, and that there is no need to estimate the ineffective area in buttress wall problems, both issues for the traditional thrust line analysis method.

Experimental Assessment of Aero-Propulsive Effects on a Large Turboprop Aircraft with Rear-Engine Installation

This paper deals with the estimation of propulsive effects for a three-lifting surface turboprop aircraft concept, with rear engine installation at the horizontal tail tips, conceived to carry up to 130 passengers. This work is focused on how the propulsive system affects the horizontal tailplane aerodynamics and, consequently, the aircraft’s static stability characteristics using wind tunnel tests. Both direct and indirect propulsive effects have been estimated. The former produces moments whose values depend on the distance from the aircraft’s centre of gravity to the thrust lines and propeller disks. The latter entails a change in the angle of attack and an increment of dynamic pressure on the tailplane. Several tests were also performed on the body-empennage configuration to investigate the propulsive effects on the aircraft’s static stability without the appearance of any aerodynamic interference phenomena, especially from the canard. The output of the experimental campaign reveals a beneficial effect of the propulsive effects on the aircraft’s longitudinal stability, with an increase in the stability margin of about 2.5% and a reduction in the directional stability derivative of about 4%, attributed to the different induced drag contributions of the two horizontal tail semi-planes. Moreover, the rolling moment coefficient experiences a greater variation due to the propulsion depending on the propeller rotation direction. The outcomes of this paper allow the enhancement of the technical readiness level for the considered aircraft, giving clear indications about the feasibility of the aircraft configuration.

Fracturing and collapse behavior of masonry vaulted structures: a lattice-discrete approach

This study presents a novel meso-scale approach to investigate the fracturing and collapse behaviors of unreinforced masonry vaulted structures induced by spreading supports. Traditionally, the behavior of masonry vaulted structures is investigated by resorting on limit analysis method, that is a limited approach as: (i) it tackles just the failure condition of the arched structural configuration as it assumes the simultaneous formation of hinges once the thrust line reaches the edge of the masonry structure, (ii) it completely ignores the damage propagation phenomenon, starting from the trigger of the first fracture up to the complete structural failure condition. The comprehension of this fracturing process is fundamental aiming to analyze intermediate damage states for the check of serviceability limit states and to individuate a more realistic structural crack distribution in ultimate conditions. This paper proposes a thorough understanding of the fracturing behavior of masonry vaults based on non-linear fracture mechanics concepts. For this purpose, the Lattice Discrete Particle Model (LDPM) is adopted to simulate a variety of stone masonry vaulted structures up to their collapse. LDPM simulates the behavior of masonry at the stone level. The interaction between stones that are bounded by weak layers of mortar is governed by specific constitutive equations describing tensile fracturing with strain-softening, cohesive and frictional shearing, and compressive response with strain-hardening. The formation of hinges, the activation of the mechanism and the kinematic mechanism are analyzed for three different types of vaults, namely groin, barrel and depressed vaults, and for six different slenderness. The first conclusion of this study is that LDPM can be used as an alternative tool to perform typical limit analysis for the assessment of safety of arches and vaults in ultimate conditions. Most importantly, LDPM is able to show that the fracturing process is a progressive phenomenon, the cracked surfaces are never strictly symmetric respect to the vertical axis and do not appear simultaneously. In particular, the features of non simultaneity and asymmetry of the cracks increases as the distance between the crown of the vault and the imposts increases, i.e. going from depressed vaults to groin vaults. Finally, the evolution of the fracturing process occurs more progressively and exhibits less pronounced asymmetry in the case of depressed vaults as compared to groin vaults for which, in turn, the damage appears to be more brittle and characterized by asymmetry in the cracks distribution.

Optimal arch shape of long-span open-spandrel arch bridges under vertical permanent loads

An arch with an optimal shape can resist loads more efficiently with higher safety. Long-span open-spandrel arch bridges need to have an optimal arch shape. This paper presents a study of obtaining the optimal arch shape of long-span open-spandrel arch bridges under vertical permanent loads. The mathematical model of the optimal arch shape was established firstly for open-spandrel arch bridges. Then the solution procedure of the mathematical model was determined. Finally, the mathematical model was applied to two arch bridge cases. The results show that the optimal shape of open-spandrel arch bridges is the linear combination of the thrust lines under both the arch self-weight and the point loads (the deck weight plus the spandrel column or hanger weight). When subjected to the arch self-weight, the arch with variable cross-section has a thrust line equation that is the total of the hyperbolic cosine function, the hyperbolic sine function, and the quadratic polynomial. As for the point loads, the corresponding thrust line is a polyline. The two long-span arch bridge cases demonstrate that the proposed mathematical model is highly accurate and efficient for practical application. The bending moments, bending stress to axial stress ratios, eccentricities, and the square sum of eccentricities generated in the optimal arch shape are significantly less than those generated in conventional arch shapes, such as parabola, catenary, polygon, and spline. The results highlight the advantages of the optimal arch shape presented in this paper over conventional arch shapes, mostly of the parabola and the catenary. The point loads are assumed to be uniformly distributed along the arch span and the arch self-weight is assumed to be uniformly distributed along the arch length in conventional approaches, which may cause significant inaccuracies in forming an arch shape. The mathematical model proposed in this paper avoids the inaccuracies caused by the above-described assumptions, and greatly enhances the calculation accuracy. As a result, the mathematical method proposed in this paper has a wide range of applications in bridge engineering.

Conceptual design of transfer structures

AbstractA transfer structure (TS) is a structure that alters the load path of the gravity loads, shifting the line of thrust laterally to a different vertical alignment. Transfer structures are introduced in buildings that feature discontinuities in some columns or walls and where a direct load path to the foundations is not possible. They usually represent major elements of the structure and their impact on building cost and construction time can be substantial. This paper presents an overview of existing transfer systems and provides guidance on their design and construction. Extensive research on buildings with transfer structures all over the world was carried out (more than 100 examples were analyzed) with the aim of developing a rational typology of these structures. The results can be broken down into five main types: BEAM, TRUSS, INCLINED STRUT, PLATE, and ARCH & TIE.

Mechanics of tubes composed of interlocking building blocks

Topologically interlocking material (TIM) systems are composed of convex polyhedral units placed such that building blocks restrict each other’s movement. Here, TIM tubes are considered as rolled monolayers of such assemblies. The deformation response of these assembled tubes under diametrical loading is considered. This investigation employs experiments on additively manufactured physical realizations and finite element analysis with contact interactions. The internal load transfer in topologically interlocking tubes is rationalized through inspection of the distribution of minimum principal stress. A thrust-line (TL) model for the deformation of topologically interlocking tubes is established. The model approximates the deformation behavior of the assembled tubes as the response of a collection of Mises trusses aligned with paths of maximum load transfer in the system. The predictions obtained with the TL-model are in good agreement with results of finite element models. Accounting for sliding between building blocks in the TL-model yields a predicted response more similar to experimental results with additively manufactured tubes.

Analysis and construction of transfer structures: a case study

A transfer structure of a building is a structure that alters the load path of the gravity loads, shifting the line of thrust laterally to a different vertical alignment. They usually represent major elements of a structure and their impact on building cost and construction time can be substantial. The design for deflection control is frequently the primary consideration when defining the geometry of a transfer element, given their usual long spans and the high magnitude of the forces involved. Considerations of robustness and disproportionate collapse may also be key for the design of transfer structures as these are often regarded as critical elements for the overall stability of the building. Furthermore, the sequence of construction should be modelled through a construction-staged analysis, as it greatly affects deflections as well as final forces distribution. This paper provides guidance and highlights the key aspects that are likely to determine the structural design of transfer structures, as well as typical construction methods and challenges. It also presents a comparative study between two different types of transfer structures for the same building: an alternative composite scheme for the post-tensioned concrete transfer grid of the St. Gabriel Tower (Lisbon) was designed and the most relevant factors are highlighted.

A Comprehensive Analysis Structure for the Design of Masonry Arches

Background: Structural masonry framing has the potential to be an advantageous design technique for new constructions. The realization of this potential requires both design modernizations and accessible analysis methodologies. The focus and understanding of masonry frames have been directed towards the management and preservation of cultural heritage. This has resulted in an assessment approach to the analysis and duality of the term “masonry” to describe both a material and a method of construction. Objectives: The objective of this work is to differentiate masonry as a method and masonry as a material and to use this differentiation to present a comprehensive method-based analysis structure for masonry arches that is formulated around the need to control and optimize the system. Methods: This work presents an analysis approach that defines and utilizes kinematic equilibrium to establish determinant systems. This is achieved through the inclusion of a loading variable to a defined mechanical condition of the arch. The solution to the equilibrium equation sets is evaluated for admissibility through the examination of the thrust line and arch geometry. The simplified analysis is formulated into a simple software structure, a first-order assessment strategy, a characterization technique to link experiment and theory, and carried to dynamic modeling. Results: The results of the approach are the foundation and blueprint for a comprehensive, efficient, and adaptable structural analysis platform designed for the structural analysis of masonry frames. Conclusion: The developed analysis approach and supporting applications cover the base requirements for promoting the application of masonry frames for new constructions.

Structural efficiency of varying-thickness regolith-based lunar arches against inertial loading

Following the current trend towards the lunar exploration and habitation, this study focuses on the development of efficient arch forms to be adopted for future long-span shielding structures on the lunar surface. More specifically, the static behaviour of the optimised, varying-thickness arches (VTAs), produced by a previously proposed iterative form-finding algorithm based on limit thrust-line analysis under gravitational and seismic loading is examined herein. These form-found arches are assumed to be constructed by laser-sintered additive manufactured lunar regolith. This paper starts with the visualisation of the limit thrust line using finite element analysis (FEA) on constant-thickness arches (CTAs). Subsequently, it presents the results from FEA on both VTAs and CTAs where it is observed that the original VTAs need to be geometrically enhanced, by thickening certain weak areas of their cross-section in order to minimise the local principal stresses and strain energy. The work proceeds with the quantification of the efficiency of the enhanced VTAs against their CTA counterparts, in both terrestrial and lunar gravitational environments. The most efficient enhanced VTAs are selected and recognised as the best structural forms for either terrestrial or lunar applications among all the arches examined in this research. Eventually, the real capacity of those most efficient arches against lateral loading is attested by means of pushover analysis. As expected, it is found that the geometric enhancement has significantly increased their structural capacity.

Masonry vaulted structures under spreading supports: Analyses of fracturing behavior and size effect

This paper deals with the fracturing behavior of unreinforced masonry arches and vaults induced by spreading supports. The traditional method of limit analysis is limited in understanding the actual failure of arches, as it assumes the simultaneous formation of hinges once the thrust line reaches the edge of the masonry structure. The damage propagation phenomenon, starting from the trigger of the fracture up to the complete structural failure is thus ignored. Moreover, limit analysis does not capture the effect of structural size on the nominal strength due to strain-softening and damage localization. This manuscript proposes a thorough understanding of the fracturing behavior and size-effect of arches and vaults based on computational modeling and non-linear fracture mechanics concepts. For this purpose, the Lattice Discrete Particle Model (LDPM) is adopted to simulate a variety of stone masonry vaulted structures up to their collapse. The formation of hinges, the activation of the mechanism and the kinematic mechanism are analyzed for three different types of vaults, namely groin, barrel and depressed vaults, and for six different slendernesses. The effect of arch size on structural strength is then analyzed using LDPM, by simulating self-similar arches of five different sizes and of three different slenderness ratios. The numerical data of size-effect is also analyzed using a newly developed analytical formula based on non-linear fracture mechanics theory and taking into account self-weight, whose effect is of paramount importance in arches and vaults under spreading supports. Results show a strong reduction of structural strength as the size increases, as a matter of fact stronger than the typically observed reduction due to energetical size-effect. The difference is due to self-weight, one of the main driving forces in the collapse of thrusting arches. This might explain the reason why in some seismic locations, small sized vaulted structures remain almost undamaged whereas larger ones often collapse.

Novel Model for Limit-Equilibrium Analysis of Slope Stability with a Nonlinear Strength Criterion

Slope stability has always been a concern of the geotechnical engineering community, and geotechnical scholars have continued to explore the reasonable failure mechanism of slopes and make the theoretical models of slope stability more reliable. This work established a novel model for limit-equilibrium (LE) analysis of slope stability with a nonlinear strength criterion. In this theoretical model, the thrust line of interslice force and corresponding increment are reasonably simplified. Then, the formula of normal stress on slip surface is derived by assuming the increment of normal interslice force. Meanwhile, strength reduction technology is applied to reduce the shear strength of the geotechnical body on the slip surface with slope factor of safety (FOS) to obtain shear stress on the slip surface. Thereafter, mechanical equilibrium conditions satisfied by the whole slope sliding body and newly introduced equation about increments of interslice force are used to solve LE stability of the slope. For stability analysis of a slope with a nonlinear strength criterion via a simple and feasible calculation process, an iterative loop solution strategy is adopted in the analysis. After comparison and analysis of several slope examples, the feasibility and rationality of the present method are verified. Moreover, the present method has high efficiency in terms of calculation speed.

Graphical Modelling of Hoop Force Distribution for Equilibrium Analysis of Masonry Domes

The behaviour of axisymmetric masonry shells can be simulated by a system of forces constituted by meridian forces acting in the vertical planes, and by hoop forces acting circumferentially. A crucial component for the assessment of these structures using the Modified Thrust Line Method (MTLM) is the determination of hoop forces, whose computation is strenuous, limiting the practical application of MTLM. Working around this limitation, the current research introduces a strategy to manipulate the hoop forces by graphically implementing a function describing their distribution. The adaptiveness of this distribution function not only allows the application of MTLM for the analysis of a range of geometries, but also enables the simulation of membrane behaviour, arch behaviour and their combination, for considering partially cracked structures. Taking this into account, the approach is applied in the case studies illustrated within the current research.

Searching for admissible thrust surfaces in axial-symmetric masonry domes: Some first explicit solutions

Determining the internal forces acting within a masonry dome or vault loaded by an assigned distribution of external actions is a problem that is still open, as evidenced by even recent contributions to the scientific literature on the topic. The present work intends to address this issue by proposing a method for determining admissible distributions of stresses in vaults and masonry domes. The problem is solved analytically with the aim of obtaining, when possible, explicit expressions for the internal forces. Although applicable in principle to arbitrarily distributed loads, the procedure adopted herein for searching for statically admissible internal forces is described in detail for the case of distributed and concentrated vertical loads. The analysis is performed by building suitable analytical solutions to the so-called “direct” and “inverse” problems of a thin shell in which bending forces are nil and only membrane forces are present. The solutions thusly obtained are applied to vaults and domes by making use of the so-called “thrust surface” concept, which represents a natural generalization of the thrust line for masonry arches. According to Heyman’s hypothesis for masonry, when the thrust surface is entirely contained within the vault thickness, a corresponding statically admissible stress field can be found. Thrust surfaces corresponding to admissible stress fields are determined by means of an expressly developed iterative procedure that begins by assigning an initial shape to the thrust surface. Then, by suitably using the solutions of the inverse problem, the shape of the thrust surface is modified so that the corresponding stresses become, when possible, statically admissible. By using the well-known theorems of limit analysis, both the mechanical and geometric safety coefficients are assessed for vaults and existing domes. As an example, the proposed procedure is applied to three practical case studies: a hemispherical dome of constant thickness, the dome of the Rome Pantheon, and the dome of Bernini’s Santa Maria Assunta Church in Ariccia (Rome).