No-tension Material Research Articles

Semianalytical Calculation of the Collapse Load of Eccentrically Compressed, No-Tension Beams with an Annular Cross Section

AbstractA second-order equilibrium analysis is developed for the calculation of the collapse load of an eccentrically compressed beam with an annular cross section in the presence of a no-tension material. Unlike the case of a rectangular cross section, extensively treated in the literature, no rigorous closed-form solution is possible, in this case, but semianalytical results can be obtained, which may be useful for engineering applications.

Limit analysis of masonry arches with finite compressive strength and externally bonded reinforcement

The collapse load of masonry arches with limited compressive strength and externally bonded reinforcement, such as FRP, is evaluated by solving the minimization problem obtained by applying the upper bound theorem of limit analysis. The arch is composed of a finite number of blocks. The nonlinearity of the problem (no-tension material, frictional sliding and crushing) is concentrated in the interface between two adjacent blocks. The crushing in the collapse mechanism is schematised by the interpenetration of the blocks with the formation of hinges at internal or boundary points of the interface. The minimization problem is solved with linear optimization, taking advantages of the robust algorithms offered by linear programming (LP). The optimal solution of the linear programming problem approximates the exact solution to any degree of accuracy. The dual of the minimization problem is also formulated and is solved in order to present the statics (thrust curve, locus of feasible internal reactions, etc.) of the reinforced arch as a consequence of the kinematical assumptions used in the primal minimization problem. Numerical examples are presented in order to show the effectiveness of the proposed method. Finally, it is shown that the results provided by the proposed LP are in good agreement with an experiment on a FRP-strengthened arch characterized by crushing failure of the masonry.

Limit analysis of FRP strengthened masonry arches via nonlinear and linear programming

The collapse load of masonry arches strengthened with FRP materials is determined. The arch is made of quadrangular blocks and the nonlinearity of the problem (no-tension material, frictional sliding and crushing) is concentrated at the interface between the blocks. Two methods are used to solve the problem. In the first method, a nonlinear programming problem (NLP) is formulated and is solved by using the successive quadratic programming algorithm (SQP) and combinatorial analysis. This method finds the optimal solution in the analysed cases. In the second method, a linear programming problem (LP) is formulated and is solved with classical techniques. LP approximates the optimal solution to any desired degree of accuracy. Although the number of variables of LP is much larger than that of NLP, LP process time can result much lower than NLP process time. Numerical examples are provided in order to show the advantages of the two methods and the effectiveness of FRP strengthening for different arch geometries.

Masonry Walls as Orthotropic No Tension Structures

In this paper, following the research line proposed by Milankovicth (1910) and Signorini (1925), a semi-analytical approach for the analysis of masonry walls - treated as horizontally multi-layered strata of no-tension material - is proposed in order to evaluate the stress distribution generated by applied loads. The main applications concern walls subjected only to vertical loads. The goal is to identify in the masonry wall the actual bearing sub-structures by defining at each vertical and horizontal level the transversal sections generated by the unilateral behaviour. The method proposed will allow identifying the partition of the wall into macro-elements, which depend on the centre of applied loads, and wall and bricks geometrical features. The technique adopted to approach the problem can be referred to as the search for the best shape. Two cases are discussed: corbelled masonry walls surrounding windows and arch behaviour of masonry over the window.

Out-of-plane seismic analysis of Bozdogan (Valens) aqueduct in Istanbul

The Bozdogan (Valens) aqueduct in Istanbul is one of the most prominent water supply structures inherited from the Byzantine period. The paper investigates the out-of-plane seismic resistance of the aqueduct. The structural system of the aqueduct is composed of a series of piers connected to each other with arches at two tier levels. Taking advantage of the structural periodicity, only one pier of the highest part of the aqueduct is considered for the analysis instead of the whole structure. This pier is modelled as a cantilever prismatic element subjected to gravity load and increasing lateral load representing out-of-plane seismic loading. It is assumed that the pier is made of a no-tension material, with a linear stress-strain relationship in compression, and has infinite compression strength. To accomplish the solution, an efficient numerical model and solution procedure developed by La Mendola and Papia for investigating the stability of masonry piers under their own weight and an eccentric top load, is utilised and adapted to the problem at hand. The analysis showed that, although, the aqueduct can withstand out-of-plane earthquake ground motions of medium size and usually encountered periods, it is vulnerable to the ones containing long-period pulses. Key words: Bozdogan (Valens) aqueduct, seismic, out-of-plane resistance, numerical model.

An Approach to Masonry Structural Analysis by the No-Tension Assumption—Part I: Material Modeling, Theoretical Setup, and Closed Form Solutions

The prevalent feature that characterizes masonry structures and makes them dissimilar from modern reinforced concrete and steel structures is quite definitely their inability to resist tensile stresses. Therefore, it is natural that the material model that is intended to be an “analog” of real masonry cannot resist tensile stress but possibly behaves elastically under pure compression, opening a perspective on the adoption of the no-tension (NT) material constitutive assumption. Founded on the NT theory, the basic direction and aim of the present work is to propose a unified treatment of masonry structures presenting an overall and comprehensive up-to-date insight in the analysis of masonry constructions while providing basic and advanced concepts and tools already available or original. The note is intended to lead to several notable results including, just to mention some of the outcomes, that the St. Venant’s postulate does not hold in NT solids, no energy is dissipated by fracture, special accommodations for discontinuous loads are needed, and the relevant developments are provided, among other significant outcomes such as the identification of operative procedures for engineering solutions of structural problems. In the first part of the paper, the basics for the foundation of a NT material theory are illustrated, and the relevant principles for structural analysis, mainly identified in the classic energy theorems and suitably adapted to the material at hand, are formulated. In (apparently) simple cases, closed-form solutions can be obtained, or, at least, the solution process can be prepared after a preliminary screening of the equilibrium scenario. The application to two-dimensional no-tension elasticity is then illustrated in the last section, with reference to two sample cases. The first example proves that the solution in a NT panel acted on by vertical loads on the top is clearly identified in terms of stress, but if discontinuities are present in the load pattern, these reflect in some strong singularities in the deformation field that requires to deepen the problem. The second example is concerned with a NT-elastic half-plane, and a technique to find approximate solutions is outlined and carried on in detail. In Part II of the note, it is demonstrated that discontinuous or lumped load patterns can be looked at as frontier cases: They are not in contrast, but in some sense they are not natural to NT solids. Still in Part II, it is shown how the equilibrium problem can be managed both through the strain and the stress approach.

An Approach to Masonry Structural Analysis by the No-Tension Assumption—Part II: Load Singularities, Numerical Implementation and Applications

The present note follows the previous one, part I, where theoretical tools for analyzing structures made by no-tension (NT) material are presented in extensive form. Some significant results are presented in this part II of the note, ranging from the highlight of problems related to the treatment of singularities possibly occurring in the modeling of loads that burden on NT structures, and the development of tools for their treatment, to the specialization of general problems to planar and spatial structures, with the related ad hoc conceived analytical set up, original numerical tools, and noncommercial software. The implementation of the problems, deriving from the set up of the constrained optimization presented in part I of the note, is developed together with a number of applications, also providing, in some cases, the comparison of numerical, experimental, and/or field evidence referred to masonry structures, which finally validates the applicability of the adopted approach for the analysis of masonry constructions.

Elastic Solutions for Masonry-like Panels

The problem of equilibrium for normal elastic no-tension materials can be formulated as a minimum problem for the complementary energy functional over the set of statically admissible stress fields ℍ. Masonry facades are often formed by an assembly of units in the form of rectangular panels. A special class of mixed boundary value problem for such units is solved by minimizing the complementary energy over the subset \(\tilde{\mathbb {H}}\) of ℍ defined by the admissible stress fields that are of rank one. These minimal uniaxial stress states are often the exact solutions of such BVP’s as verified by deriving the corresponding fracture strains through the compatibility equations and checking the boundary conditions for the associated displacements. The novelty of the proposed approach, besides the generality of the formulation leading to the uniaxial stress solution (i.e., the compression ray distribution) in all the possible combinations of given rigid displacements of the bases, resides in the analytical description of the associated anelastic strain.

Equilibrated divergence measure stress tensor fields for heavy masonry bodies

Abstract In this paper we study the equilibrium problem of rectangular panels made of no-tension material under gravity and undergoing a vertical load distributed on their top and a horizontal load distributed on a lateral side. Equilibrated and negative semi-definite stress fields are presented which are tensor-valued measure whose distributional divergence is a vector-valued measure. Then, for a fixed vertical load, the maximum value of the lateral load which is compatible with the equilibrium of the panel is determined.

Integration of measures and admissible stress fields for masonry bodies

We study the compatibility of loads for bodies made of a no-tension (masonry) material. Loads are defined as weakly compatible if they can be equilibrated by an admissible stress field represented by a tensor valued measure, and strongly compatible if they can be equilibrated by a square integrable function. In the present study, we examine situations in which weak compatibility implies strong compatibility. For families of loads that depend on a parameter and the families of measures that equilibrate these loads, we find that, under some conditions, averaging with respect to the parameter leads to a measure with a square integrable density that equilibrates the loads. We illustrate the procedure on twodimensional rectangular panels free from gravity, clamped at the bottom, and subjected to various loads on the free part of the boundary.

A note on equilibrated stress fields for no-tension bodies under gravity

We study the equilibrium problem for two-dimensional bodies made of a no-tension material under gravity, subjected to distributed or concentrated loads on their boundary. Admissible and equilibrated stress fields are interpreted as tensor-valued measures with distributional divergence represented by a vector-valued measure, as developed by the authors of the present paper. Such stress fields allow us to consider stress concentrations on surfaces and lines. Working inRn,\mathbb {R}^n,we calculate the weak divergence of a stress field that is asymptotically of the form|x|−n+1T0(x/|x|)|\mathbfit {x} |^{-n+1}\mathbfit {T} _0(\mathbfit {x} /|\mathbfit {x} |)forx→0\mathbfit {x} \to \mathbf {0}on a region that is asymptotically a cone with vertex0\mathbf {0}. Such stress fields arise as parts of our solutions for two-dimensional panels. Proceeding to problems in dimension two, we first determine an admissible equilibrated solution for a half-plane under gravity that underlies two subsequent solutions for rectangular panels. For the latter we give solutions for three types of loads.

No tension beam-columns with bounded compressive strength and deformability undergoing eccentric vertical loads

This work concerns the static behaviour of rectangular cross-section beam-columns made of a no-tension material with limited compressive strength and deformability and subjected to eccentric vertical loads acting at their extremities. The explicitly obtained results allow for thorough description of the stability characteristics of such a pile. A final example shows the practical potentials of the method.

Relationships of LA theorems for NRT structures by means of duality

The paper refers to previous works developed by the authors, dealing with the possibility of applying duality theorems to non-linear programs coming out from limit analysis (LA) of structures made by not resisting tension (NRT) or no-tension material. Under such perspective, after setting up the static or kinematic LA problem for NRT structures, the main task, for duality theorems to be applicable, is to demonstrate some convexity-related properties of the involved functions and domains. This feature, which is of basic importance for the whole procedure, is not trivial since all of the required conditions are to be accurately checked by analytical developments. Application of duality is finally demonstrated to give a complete and clear interpretation from a physical point of view about relationships relevant to LA approaches.

LONGITUDINAL OSCILLATIONS OF BIMODULAR RODS

In this paper we consider the longitudinal vibrations of bimodular rods. After proving the equivalence between the entropy condition and Lax condition (whenever this latter is applicable), we go on to consider the Riemann problem and prove that a unique solution always exists except for the special case of no-tension materials. This procedure allows for easy determination of the explicit solution.

A numerical method for the limit analysis of masonry structures

The paper presents a numerical method for the limit analysis of structures made of a rigid no-tension material. Firstly, we formulate the constrained minimum problem resulting from the application of the kinematic theorem, which characterizes the collapse multiplier as the minimum of all kinematically admissible multipliers. Subsequently, by using the finite element method, we derive the corresponding discrete minimum problem in which the objective function is linear and the inequality constraints are linear as well as quadratic. The method is then applied to some examples for which the collapse multiplier and a collapse mechanism are explicitly known. Lastly, the solution to the minimum problem calculated via numerical codes for quadratic programming problems, is compared to the exact solution.

Explicit Solutions for the Stability of No-Tension Beam-Columns

This work concerns the stability of rectangular cross-sectional piles made of a no-tension material and subjected to an axial load acting at the extremities within the middle third of the cross section. The resulting differential equations are solved, and an explicit relation between the load and a suitable deformability parameter obtained.

Stability of columns with no tension strength and bounded compressive strength and deformability. Part I: large eccentricity

This work concerns the stability of piles with rectangular cross section, made of a no-tension material with limited compressive strength and deformability, subjected to an axial load acting within the cross section but outside its middle third (large eccentricity). The differential equations obtained have been solved explicitly, and their solutions have allowed us to describe the stability characteristics of the pile through graphical representations.

A numerical method for solving equilibrium problems of no-tension solids subjected to thermal loads

This paper starts out by recalling a constitutive equation of no-tension materials that accounts for thermal dilatation and the temperature dependence of the material parameters. Subsequently, a numerical method is presented for solving, via the finite element method, equilibrium problems of no-tension solids subjected to thermal loads. Finally, three examples are solved and discussed: a spherical container subjected to two uniform radial pressures and a steady temperature distribution, a masonry arch subjected to a uniform temperature distribution and a converter used in the steel and iron industry.

Thermodynamics of no-tension materials

This paper presents a constitutive equation for no-tension materials in the presence of thermal expansion that accounts for the temperature-dependence of their material’s constants. Specifically, assuming that the symmetric part of the displacement gradient minus the thermal dilatation is small, an explicit expression is given for stress from which the free energy, internal energy, entropy and enthalpy are obtained. Then, the equilibrium energy equations of a no-tension solid are presented, and we observe that, under further suitable hypotheses, thermo-mechanical uncoupling occurs. Finally, the work lost during an adiabatic process by two spherical containers made of a linear elastic and a no-tension material are compared.

Limit analysis and no-tension materials

The applicability of the two fundamental theorems of Limit Analysis outside the domain of Perfect Plasticity, in which they were originally established, is discussed. Specifically, a class of materials called normal linear elastic, including no-tension materials as a particular case, is considered, and it is proved that the theorems of Limit Analysis do apply to this class. The proof is based on the circumstance that perfectly plastic materials and normal linear elastic materials admit the same characterization of collapse.