This paper presents an application of vibration analysis to the monitoring of tierods. An algorithm for the axial load estimation based on experimentally measured natural frequencies is introduced and its application to a case study is reported. The proposed model of a tierod incorporates elastic bedtype boundary conditions that represent the contact between stonework and the tierod. The weighed differences between experimentally and numerically determined frequencies are minimized with respect to the parameters of the model, the main being the axial load and the stiffness at the tierod/wall interface. Thus, the multidimensional optimization problem is solved. Results are analysed in comparison to a model with simple fixedend boundary conditions. In addition, the analytical formulation of the problem is delivered.
The present paper reviews the applications of vibration analysis to the monitoring of the socalled
Tierods in the Castle of Torrechiara in Langhirano, Parma, Italy.
Over the years, deformations of masonry walls and eventual displacements in the building may cause significant changes in the axial loads of tierods. In the extremes, this can lead to either of two scenarios: failure in structural integrity of tierods (damages and cracks) or loss of loads and subsequent performance decline, a phenomenon referred to as the “laziness” of tierods. Both of the scenarios are dangerous for the safety and integrity of buildings and can lead to irretrievable harm to the precious historical heritage of the human race. For this reason regular monitoring of tierods’ condition is of a great importance.
Health monitoring of tierods includes two major steps. The first one is identification of axial load and the second one is damage identification. As for the first one, multiple methods have been developed to accomplish this task and the details on the state of art are provided in the next section. Such experimental techniques should be as less invasive as possible and at the same time provide sufficient data on the beam condition. Generally this type of testing is referred to as “nondestructive.”
In particular, nondestructive testing (NDT) is the process of investigating structures and elements for characteristics, discontinuities, changes in properties, and so forth without harming the continuity and usability of the part under testing. One of the relatively cheap, easily executable, and reliable NDT techniques is vibration analysis (VA). This way of health monitoring of the structures can be applied to testing whole buildings as well as its smallest parts depending on the scope and approach used. VA is based on investigation of dynamics of a structure under a certain excitation: it can be an impact hammer or a shaker. The response to the excitation is registered via sensors: accelerometers, optic sensors, laser, and so forth. Vibrational response contains information about the main structural characteristics of the system: mass and stiffness. Based on the knowledge of modal parameters, conclusions are drawn on the loading and structural integrity of the elements.
A reliable experimental technique helps to validate analytical and numerical models used for prediction of cracked beam dynamics. The purpose of this research is to develop a VA procedure based on quantitative and qualitative analysis of frequency response functions.
The further sections describe a method for axial load identification in tierods developed by the Department of Industrial Engineering of the University of Parma. This approach combines in situ dynamic tests and computations that make use of a beam model with complex boundary conditions. The method was tested and improved throughout some years since it was applied for multiple case studies of monitoring such famous Italian historic buildings as
The structural characterization of tierods is crucial for the safety assessment of historical buildings. The main parameters that characterize the behaviour of tierods are the tensile force, the modulus of elasticity of the material, and the rotational stiffness at both restraints. In the last decades several techniques for an indirect nondestructive evaluation of such parameters have been proposed. The nondestructive procedures currently available for the structural characterization of tierods can be grouped in static, staticdynamic, and pure dynamic approaches. Pioneering static methods presented, for instance, in works of Pozzati [
Mixed approaches try to identify the unknown parameters by combining static and dynamic measures. Blasi and Sorace [
Such drawbacks are avoided in pure dynamic procedures [
Lagomarsino and Calderini [
Recently Maes et al. introduced a method that enables definition of axial loads in slender beams with unknown boundary conditions, taking into account effects of rotational inertia of the beam and masses of sensors [
Rebecchi et al. established an analytical method of processing experimental data from five instrumented sections of a prismatic slender beam, which showed excellent results in estimation of the axial load in tierods [
Livingston et al. identified the tensile force in prismatic beams of uniform section by using modal data and assuming rotational and vertical springs at each end of the beam [
Another fully dynamic procedure has been proposed by Kim and Park [
Amabili et al. [
The first step of the method is the in situ experimental identification of natural frequencies of the tierods by measuring the frequency response functions (FRFs) via instrumented hammer excitation. Precisely the testing technique used for the investigation of tierods in Casa Romei located in the city of Ferrara, EmiliaRomagna, Italy, was described in [
The interface tierod/wall was assumed to be a continuous elastic bed; that is, extremities of tierods inserted inside masonry walls were modelled as resting on Winklertype foundation. This type of boundary has been used in dynamics of particular cases, for example, for beams or rails subjected to travelling loads, as reported by Farghaly and Zeid [
Tierod with elastic bedtype boundaries.
Clearly, the foundation may have a nonuniformly distributed stiffness, which would result in different
As shown in Figure
Assuming the hypotheses of BernoulliEuler beam theory for the analytical formulation, we chose to neglect the shear deformation and rotational inertia, because the subject of this study was a slender rod, for which the ratio of linear dimensions of crosssections to length is a very small number.
The energy approach was used to obtain the equations of motion via Lagrangian of the system [
Kinetic energy is
We proceeded defining the Lagrangian of the system, which is equal to the difference between the overall kinetic and potential energy:
Tierods were modelled in FEM software using beam elements. In this case the beam is represented with a onedimension body, that is, a wire (line, curve, polyline), and crosssection shapes and dimensions are assigned to this body as one of the properties, which allowed taking into account irregular crosssections, added masses, elastic supports, and so forth. The beam model shows good results for analyses of long slender beams. For beam elements there is an option of modelling the axial tensile load as a bolt pretension load. Tierods were modelled using 50–60 B31 2node linear beam elements in 3D space implying Timoshenko’s beam theory.
The FEM simulation was divided into two steps: as a first step a pretension load
As a first iteration we analytically investigated the function
Subsequently we modelled the elastic bed boundaries, representing a general condition of translational and rotational stiffness acting for the length
forms a matrix of parameters;
launches the FEM analysis for each nod of the grid, extracts and filters natural frequencies from the output;
calculates the value of residual error (
finds the local minimum of the function (
refines the grid of parameters and repeats the procedure again.
The method described hereby was applied to investigation of tierods installed in “Casa Romei” located in Ferrara, Italy (see Figures
Inner yard of Casa Romei and zoom of the ground floor tierods.
Our team during hammer excitation of a tierod; map of ground floor.
First, measurements of geometrical characteristics and of natural frequencies were performed for each tierod. Then experimental acquisition was carried out to determine the natural frequencies from the analysis of response to dynamical excitation applied to tierods in horizontal plane; as an example, a frequency response function (FRF) for a ground floor tierod is shown in Figure
FRF plot (acceleration amplitude versus frequency).
For further analysis, first four to six natural frequencies were identified for each tierod. Six eigenmodes were considered sufficient, since identification of higher modes might appear inaccurate due to larger possible measurement errors.
Having performed multiple experimental studies of tierods, the authors concluded that the variation of the material was less significant than in the boundary conditions. Thus, the material properties were kept constant in this case study: material was assumed to be general iron with characteristics:
Table
Experimental acquisitions.
Tierod number  Crosssection 
Length 
Natural frequencies [Hz]  

I  II  III  IV  V  VI  
PT1 

3178  15.30  —  49.00  —  92.30  — 
PT2 

3233  16.80  —  53.80  —  98.80  — 
PT3 

3228  16.30  —  53.80  —  103.00  — 
PT4 

3218  16.00  33.50  51.30  71.80  95.00  121.80 
PT5 

3188  5.50  —  28.30  —  65.50  91.00 
PT6 

2748  21.75  49.00  85.25  131.20  188.50  255.20 
PT7 

2768  20.00  45.50  80.00  123.80  179.20  242.20 
PT8 

3358  14.75  32.75  55.75  85.25  121.20  176.50 
PT9 

3248  15.50  34.25  58.75  90.00  128.00  173.00 
PT10 

3388  17.75  38.00  63.25  94.75  133.00  177.50 
PT11 

3440  13.25  29.75  51.25  78.25  111.50  150.80 
PT12 

3298  14.50  30.25  48.25  69.25  94.00  122.20 
PT13 

3140  13.75  —  47.25  —  95.75  — 
PT14 

2510  19.25  46.50  84.00  —  134.50 
A typical graphic scenario of the optimization process is depicted in Figure
Residual error function as function of elastic bed stiffness and axial load.
Naturally, the sodefined minima of the residual error (
The model with elastic foundation boundaries delivered improvement of results as illustrated in Figure
Influence of the elastic bed on the axial load determination.
Table
Summary of results.
Tierod number  Axial load 
Stress [MPa]  Bed stiffness 
Residual error 

PT1  29.40  62.82  50.00  0.31 
PT2  46.90  102.18  0.48  6.04 
PT3  37.50  72.12  253.00  3.95 
PT4  38.70  75.88  3.75  0.77 
PT5  1.00  1.89  26.75  1.17 
PT6  66.50  66.50  113.00  0.48 
PT7  54.50  54.50  57.50  1.43 
PT8  46.30  46.30  60.00  6.60 
PT9  44.50  44.50  62.50  7.33 
PT10  79.90  79.90  76.00  2.22 
PT11  30.10  50.17  99.50  6.53 
PT12  37.20  62.00  4.25  0.48 
PT13  28.20  47.00  5.50  0.27 
PT14  38.00  63.33  2.13  5.58 
Tensile loads cause normal stresses that need to be estimated for evaluation of reliability and integrity of tierods. This safety assessment can be carried out based on the values of average axial stress
In Figure
Average stress state in tierods: comparison between encastré and elastic foundation boundaries.
In Section
A sensitivity analysis has been conducted to evaluate the influence of weight coefficients on our results. The plot in Figure
Results of computation with three sets of weights for the tierod PT4.
Sets of weight coefficients  Axial load 
Bed stiffness 
Natural frequencies [Hz]  

I  II  III  IV  V  VI  

39.00  3.50  16.03  32.91  51.37  72.03  95.32  121.53 

38.60  4.50  16.04  32.95  51.48  72.25  95.69  122.13 

38.70  3.75  16.00  32.85  51.31  71.97  95.28  121.55 

32.20  ∞  15.68  32.47  51.33  72.97  97.91  126.50 
Experimentally defined frequencies  16.00  33.50  51.30  71.80  95.00  121.80 
Residual error functions for sets of weight coefficients
In this paper a procedure for axial load identification in structural tierods was demonstrated and approved via an experimental study of an ancient mansion. The method is based on a tierod model represented by a beam with ends supported by an elastic Winklertype foundation. The elastic bed was used to simulate the contact condition between a tierod and a masonry wall. The proposed method consisted of an experimental and a computational stage. The experimental part was a relatively simple vibrational test for natural frequencies identification. The computational part was an optimization procedure for axial load estimation based on finite element modelling. The optimization has been done with respect to two parameters: the sought axial load and the distributed stiffness of the elastic bed at the boundaries. The technique provided a solution for uncertain boundary conditions and is capable of identifying axial load with high accuracy.
Investigation of the behaviour of natural frequencies depending on the parameters showed that axial load tends to shift the set of frequencies (the higher the load the higher the frequencies), while the elastic foundation stiffness changes the “distance” between natural frequencies.
As a result, consideration of elasticity at anchorages exhibited increase in axial load by up to 40%. This means that assumption of simple boundary conditions is not sufficient for modelling a tierod dynamic response. The sensitivity analysis has proved that the optimization result was stable to variation of weight coefficients and was converging to the same axial load. Thus, the method has been approved in practice and is suitable for in situ identification of axial load in ancient tierods.
Crosssection area of the tierod
Coefficients in the time function of the solution for tierod deflection
Coefficient of the general solution for the form function
Coefficients of the solution for tierod deflection that satisfy the specific boundary conditions
Constant equal to the ratio
Elastic modulus of the tierod material
Axial load acting on the tierod
Natural frequency number
Natural frequency number
Heaviside function
Moment of inertia of the crosssection about
Distributed stiffness of the elastic bed
Stiffness of the separate springs that discretize the elastic bed
Real constants used in the expressions for
Total length of the tierod
Lengths of the tierod portion inserted into the wall, sections I and III
“Free” length of the tierod, section II
Matrix of the system of linear algebraic equations for the unknowns
Poisson’s ratio
Total potential energy
Weight coefficient of a frequency number
Residual error between two sets of
Density of the tierod material
Coefficients in the exponent of the solution for tierod deflection
Kinetic energy
Time function in the Fourier solution for tierod deflection
Potential energy of elastic strain
Potential energy associated with the axial load
Potential energy associated with the elastic foundation
Natural frequency in rad/s
Tierod deflection in
Deflection for each section of the tierod
Form function of the tierod deflection
Form functions for each section of the tierod.
The authors declare that there are no conflicts of interest regarding the publication of this paper.