# Modal analysis of a riveted lattice beam for improved retrofitting operations

Hannah Franz, Sylvain Chataigner, Lamine Dieng, Mario Rinke

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Hannah Franz, Sylvain Chataigner, Lamine Dieng, Mario Rinke. Modal analysis of a riveted lattice beam for improved retrofitting operations. Eurosteel, Sep 2023, Amsterdam, France. pp.333-338, 10.1002/cepa.2279. hal-04472865

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# Modal analysis of a riveted lattice beam for improved retrofitting operations

Hannah Franz<fnref n="1" /> | Sylvain Chataigner<fnref n="1" /> | Lamine Dieng<fnref n="1" /> | Mario Rinke<fnref n="2" />

## Correspondence

M.Sc. Hannah Franz
Université Gustave Eiffel,
MAST/SMC
Campus de Nantes
Allée des Ponts et Chaussées
CS4
44344 Bouguenais Cedex
France
Email: hannah.franz@univ-eiffel.fr

## Abstract

Wrought-iron and mild-steel riveted lattice beams are well-represented in the metallic construction heritage of 1850-1930. Many rafters and purlins of roof structures of train sheds or market halls are made of flat and angle bars assembled with single-riveted joints. During retrofitting operations, those beams are often reinforced to prevent them from buckling. However, the assessment of the buckling risk depends strongly on the modelling assumptions regarding the out-of-plane rotational stiffness of the riveted joints. This paper therefore presents an experimental and numerical investigation aiming at quantifying this joint stiffness. A 3m-long riveted lattice beam, dating from 1930, was collected from a demolition site. An experimental modal analysis was carried out and the measured natural frequencies and mode shapes were used to calibrate a finite-element model of the beam. A parametric study suggests that the model with clamped connections, i.e. with high out-of-plane rotational stiffness, is in better agreement with the experimental results. This result could lead to a less conservative buckling risk assessment and less extensive strengthening measures.

## Keywords

Lattice beams, Riveted joints, Rotational stiffness, Modal analysis, Buckling, Retrofitting

# 1 Introduction

Metallic constructions of the period 1850-1930 are an important part of the French architectural heritage. Many of these constructions, such as railway bridges or train sheds, keep serving their original purpose. For them to meet current safety and durability requirements, they need to be regularly renovated. A refined assessment of their structural performance is a key to sensitive and targeted retrofitting operations.

Roof structures of historic train sheds or market halls are widely constituted of riveted lattice beams, made of plates and angles assembled using single-riveted joints. In renovations of the last twenty years, lattice purlins or rafters have often been reinforced because of numerically identified stability problems. Buckling issues can concern both the web members and the chords of the lattice beams (Fig. 1). In reality, excessive deformations rarely occur.

The assessment of the buckling risk depends strongly on the modelling assumptions regarding the out-of-plane rotational stiffness of the riveted joints [1]. Recent experimental studies from the literature have characterized the strength of riveted joints, without considering their stiffness [2-4].

Numerical studies have focused on the stiffness of certain types of riveted joints, taking into account the clamping force resulting from the riveting process [5-7]. An experimental characterisation of the rotational stiffness of riveted joints, especially single-riveted ones, is still missing.

<figure><img src="image_2.png" /><figcaption>**Figure 1** Schematic view of a lattice beam with single-riveted joints (a) in elevation and (b) in cross-section. Possible buckling shapes: (a) buckling of a web member or (b) buckling of a chord.</figcaption></figure>

This paper therefore presents an experimental and numerical investigation aiming at assessing this property, based on the modal analysis of a riveted lattice beam. In the literature, modal analysis already proved to be relevant for the identification of the rotational stiffness of connections

<footnote n="1">¹ Université Gustave Eiffel,
MAST/SMC, Bouguenais, France</footnote>

<footnote n="2">² Faculty of Design Sciences, Uni-
versity of Antwerp, Antwerp, Bel-
gium</footnote>
of existing structures [8, 9]. The experimental modal analysis carried out in this paper consists of measuring the structure's vibration properties in order to validate or update a finite element model of the beam. This paper explores which outputs of the modal analysis are the best indicators for the rotational stiffness, by comparing the experimental results with the numerical model.

## 2 Description of the lattice beam

The studied riveted metallic lattice girder was collected from a demolition site. It belonged to the roof structure of platform sheds, built in 1930 at the Gare de l'Est in Paris, from which a small part was dismantled in 2021 (Fig. 2). It is a 3,3m long Pratt-type cantilever, with a height varying from 19cm at its tip to 34cm at the support (Fig. 3). The bar sections were given by original plans and verified by measurements with a calliper. The chords are made of double L45x45x5 angles, while the web members are made of 40x7 flat bars. The diagonal and vertical web members are connected to the chords by single rivets of 12mm diameter. The beam is made of mild steel.

<figure><img src="image_3.png" /><figcaption>**Figure 2** Original setting of the studied lattice beam: the platform sheds of the station Paris Gare de l'Est. Source: SNCF-AREP.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3** Elevation drawing of the tested lattice girder.</figcaption></figure>

## 3 Experimental modal analysis

### 3.1 Experimental setup

The original support conditions of the cantilever were reproduced in the laboratory. The structure was excited using an impulse hammer with an extra soft tip, resulting in a frequency range of 150 Hz. To retrieve the structure's response, 12 capacitive accelerometers, 7 single-axis and 5 tri-axis, of sensitivities between 100 and 1000 mV/g were available. They could easily be fixed to the beam with magnets glued on 3D-printed plastic supports, designed to fit the accelerometers (Fig. 4).

To obtain satisfactory mode shapes, 30 measurement points were defined, 11 on the top chord, 11 on the bottom chord and 8 on distinct web members. Measurements of modal displacements were carried out only in the out-of-plane direction. Three sensor setups were defined (Fig. 5):

1. A "top chord" (TC) setup with 11 accelerometers on

the top chord and 1 accelerometer on the bottom chord.

2. A "bottom chord" (BC) setup with 11 accelerometers on the bottom chord and 1 accelerometer on the top chord.

3. A "web members" (WEB) setup with 8 web members equipped with 1 accelerometer at the middle and 4 accelerometers on the top and bottom chords.

The two accelerometers at points 11 and 14 were present in all setups, to be used as references. In each setup, 7 identical tests were carried out. The beam was excited with one horizontal impulse at the tip of the top chord. The signals were recorded with a frequency of acquisition of 4800 Hz during 2.5 seconds. After this time, out-of-plane accelerations decayed to near zero. The resulting sampling resolution was 0.4 Hz. Data acquisition was performed with the HBM Quantum X system, with a module MX1615B and a module MX1601B, and the associated software catman [10].

<figure><img src="image_5.png" /><figcaption>**Figure 4** View of the instrumented beam in the laboratory.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 5** Sensor setups: (a) "TC" setup; (b) "BC" setup; (c) "WEB" setup.</figcaption></figure>
## 3.2 Digital signal processing

Test outputs were analysed to identify the modal parameters, which are the coefficients of the theoretical expressions of the frequency response functions best-fitting experimental results. A frequency response function (FRF) represents the ratio between a harmonic response $X_j$ in one degree of freedom j and the corresponding harmonic excitation force $F_k$ applied at a degree of freedom k:

$$ \alpha_{jk}(\omega) = \frac{X_j}{F_k} \quad (1) $$

For a system with N degrees of freedom, that is with N modes, any FRF element can be expressed using equation (2):

$$ \alpha_{jk}(\omega) = \sum_{r=1}^{N} \frac{r A_{jk}}{\omega_r^2 - \omega^2 + i \eta_r \omega_r^2} \quad (2) $$

Where $\omega_r$, $\eta_r$ and $rA_{jk}$ are respectively the natural eigenfrequency, the damping loss factor and the modal constant of mode r. They are the modal parameters to be extracted. The modal constant is a complex number whose modulus is the amplitude of the FRF and whose angle is the phase.

For each test, the excitation and response signals were transformed from the time domain to the frequency domain using Fast Fourier Transformation (FFT). The power spectral density $S_{ff}$ of the excitation and the cross-spectral density of excitation and response $S_{xf}$ were then computed as a multiplication of Fourier transform and Fourier transform conjugates:

$$ S_{ff}(\omega) = F(\omega) \cdot F^*(\omega) \quad (3) $$

$$ S_{xf}(\omega) = X(\omega) \cdot F^*(\omega) \quad (4) $$

Where $F(\omega)$ and $X(\omega)$ are the FFT transforms of the excitation and the response signal, and $F^*(\omega)$ and $X^*(\omega)$ their respective conjugates. As each test was repeated 7 times, the spectral densities were averaged out. They were then used to calculate an estimator of the frequency response function:

$$ H(\omega) = \frac{S_{xf}}{S_{ff}} \quad (5) $$

An example of FRF is presented in Fig. 6.

<figure><img src="image_7.png" /><figcaption>Figure 6 Inertance magnitude in m/s²/N at point 17 in the “BC” setup.</figcaption></figure>

Modal parameters were extracted using the “line-fit” method [11], through an adapted existing Matlab code [12]. It is a curve-fitting procedure matching the theoretical expression of individual FRFs (precisely, the receptance) with the measured data. It performs the analysis in the frequency domain for a single mode at a time (single-degree of freedom “SDOF” method). This method is valid only if eigenfrequencies are well-separated in the considered frequency range. Indeed, the line-fit method, like all SDOF methods, relies on the assumption that in the vicinity of a resonance, the behaviour of the system is dominated by a single mode.

## 3.3 Results

Modal parameters were extracted for the first six modes observed on the FRFs. Their eigenfrequencies in Hz, with the corresponding standard deviations, are presented in Table 1. The standard deviations were obtained from the measured eigenfrequencies for each accelerometer in all three setups. These allowed obtaining 36 measurements in total.

Regarding the mode shapes, the “TC” setup was defined as the reference setup. The mode shapes of the two other setups were re-scaled using the mode shape’s amplitude of the two reference accelerometers, 11 and 14, in the reference setup. All mode shapes could then be merged together [13].

<table><caption>Table 1 Mean eigenfrequencies of the first six modes identified by modal testing, with their standard deviation (SD).</caption><thead><tr><th>Experimental mode number</th><th>f [Hz]</th><th>SD [Hz]</th></tr></thead><tbody><tr><td>1</td><td>3.37</td><td>0.12</td></tr><tr><td>2</td><td>10.82</td><td>0.37</td></tr><tr><td>3</td><td>26.17</td><td>0.10</td></tr><tr><td>4</td><td>42.92</td><td>0.06</td></tr><tr><td>5</td><td>74.09</td><td>0.20</td></tr><tr><td>6</td><td>97.29</td><td>0.24</td></tr></tbody></table>

# 4 Finite element model

## 4.1 Element types

The finite element model was created as a hybrid structure, including beam and shell elements, with the software Marc [14]. The linear chords and web members were modelled with beam elements, while the gussets at both ends and the vertical angles at the support were modelled with shell elements (Fig. 7(a)).

## 4.2 Boundary conditions

At the support, the displacements of the surfaces of the vertical angles facing the wall were fixed, probably resulting in a more rigid support condition than in reality. At the connection between chords and web members, two distinct nodes were preserved. Local coordinate systems were defined at each connection, with the z-axis going along the web member, and the x-axis pointing in the out-of-plane direction (Fig. 7(b)). The connecting nodes were
tied for the first three DOFs, corresponding to displacements in directions x, y, and z, and DOF 6 corresponding to the in-plane rotation. Regarding DOF 4, corresponding to the torsion of web members, the nodes were tied only at one end of each web member. The coupling condition for DOF 5 was the parameter of interest, controlling the out-of-plane rotation. To model a hinged connection, no coupling condition was defined. To model a clamped connection, the nodes were tied for this DOF. To embrace the full spectrum between a hinged and a clamped connection, the connection was modelled with a spring of varying stiffness $k_r$. The range of relevant stiffness values was defined based on the stiffness index $R$ [15]:

$$R = k_r \frac{L}{EI} \quad (6)$$

Where $L$ and $EI$ are the length and the rigidity of the web member. To model a near-hinged connection, $R$ must be small, while to model a near-clamped connection, $R$ must be large. Stiffness values were therefore chosen so that $R$ varied approximately between 0,01 and 100. Even though web members had varying lengths, the same rotational stiffness $k_r$ was assigned to all connections.

<figure><img src="image_8.png" /><figcaption>Figure 7 (a) View of the finite element model; (b) Detail of local coordinate systems at the connections.</figcaption></figure>

## 4.3 Material properties

As the beam was made of mild steel, the volumic mass was supposed to be about 7850 kg/m³. However, to take into account the weight of the rivets, the accelerometers and the 1mm-thick layer of paint, an equivalent volumic mass of 8450 kg/m³ was considered. The Young modulus was taken equal to 200 GPa.

# 5 Comparison of experimental and numerical results

## 5.1 Reference model

The experimental results are first be compared with the modal properties resulting from the numerical model with connection nodes tied for all degrees of freedom, taken as reference model. In the frequency range of interest, this "clamped" model yielded 7 modes: the 6 out-of-plane modes that were experimentally measured, and 1 "vertical" in-plane mode, that could not be accounted for with

the presented experimental setups. Fig. 8 presents a comparison of numerical and experimental eigenfrequencies and mode shapes, only for the out-of-plane modes. The numerical eigenfrequencies were higher than the experimental ones, as expected for this model with very stiff boundary conditions.

The mode shapes shown in Fig. 8 are equivalent to a top view of the deformed structure, as only out-of-plane displacements along the global horizontal axis are presented. Mode shapes were normalized following this procedure:

1. The experimental mode shapes were first re-scaled so that the maximum amplitude of the top or the bottom chord was equal to 1.

2. The numerical mode shapes were then re-scaled so as to obtain the best fit between the numerical and experimental mode shapes of the chords (least-square method).

The mode shapes were found to be in very good agreement for the first four modes. Real discrepancies appeared for modes 5 and 6. In Fig. 8, the numerical median between top and bottom chords' displacements is plotted as a dotted line. It enhances that in modes 5 and 6, the numerical mode shapes exhibited larger web member deformations. In the experimental mode shapes, web member deformations remained closer to the median. Fig. 9 provides a visualization of the numerical mode shape of mode 5.

<figure><img src="image_9.png" /><figcaption>Figure 8 Comparison of numerical and experimental mode shapes (out-of-plane displacements plotted along global horizontal axis).</figcaption></figure>
## 5.2 Parametric study on joint stiffness $k_r$

A parametric study was conducted to evaluate the impact of the rotational joint stiffness $k_r$ on the numerical modal properties. Table 2 presents the obtained frequencies for different values of joint stiffness. The maximum considered frequency was chosen as the frequency of the 6th out-of-plane mode of the “clamped” model presented in the previous section (f=106.15 Hz). As reducing $k_r$ from the infinite (perfectly “clamped” connection) to zero (perfectly “hinged” connection) decreased the overall rigidity of the system, the expected frequencies corresponding to the experimental modes had to be lower than 106.15 Hz.

<figure><img src="image_10.png" /><figcaption>**Figure 9** Deformed and original states of the structure for the 5th mode, obtained with the “clamped” numerical model (reference model): (a) 3d-view; (b) top view. The colours correspond to different quantities of out-of-plane displacements along z.</figcaption></figure>

Considering the frequencies of the first out-of-plane modes, it seems from Table 2 that the model with hinged connections provides a better agreement with the measured frequencies presented in Table 1. However, it must be reminded that the support conditions of the numerical model were stiffer than reality, potentially yielding higher frequencies. The frequencies of the first modes are therefore not considered as a reliable indicator in this study.

The numerical and experimental mode shapes of the out-of-plane modes 1 to 6 were quantitatively compared through a quality criteria proposed in [8]. For each mode $r$, this quantity measures the gap between the numerical and the experimental mode shapes, based on the MAC matrix (“Modal Assurance Criterion”) [11]:

$$L_r = 1 - MAC_r = 1 - \frac{|\Phi'_{r,num} \cdot \Phi_{r,exp}|^2}{(\Phi'_{r,num} \cdot \Phi_{r,num}) \cdot (\Phi'_{r,exp} \cdot \Phi_{r,exp})} \quad (7)$$

Where $\Phi_r$ is the eigenvector containing the mode shape deformations of mode $r$ at the 30 measurement points of the three combined experimental setups. $MAC_r$ is the sum of the diagonal elements of the MAC matrix. The *num* and *exp* indices correspond to the values obtained numerically and experimentally. $L_r$ varies between 0 and 1. The closest $L_r$ is to 0, the better the agreement between experimental and numerical mode shapes is. Table 3 presents the values calculated for $L_r$. Those values decrease when the stiffness increases. This result suggests that the model with clamped connections best represents the experiments. The increase of $L_r$ when $k_r$ decreases was mostly effected due to larger web deformations in the numerical model. As

an example, the numerical mode shape of mode 5 obtained with the “hinged” model is presented in Fig. 10. From the comparison with Fig. 9, it can easily be noted that the web deformations become out-of-proportion.

Moreover, Table 2 evidences one important effect of lowering the out-of-plane rotational joint stiffness: in the models with lower stiffness $k_r$, a number of additional modes appeared. Those modes were dominated by the deformations of one or several web members. Despite measurement points being located on web members in the “WEB” setup, those modes were not observed experimentally. The horizontal dashed line in Table 2 shows the limit above which it was not possible to attribute numerical modes to the experimental ones. For example, in the model with $k_r = 10E6$ Nmm, the modes 7 and 8 had close frequencies and mode shapes, and they were not in satisfactory agreement with the experimental 6th mode. This observation confirms that the models with higher stiffness $k_r$ are in better agreement with the experiments.

<table><caption>**Table 2** Eigenfrequencies of the numerical modes in the frequency range [0-106.15] Hz.</caption><thead><tr><th rowspan="2">Numerical mode number</th><th colspan="5">k<sub>r</sub> [Nmm]</th></tr><tr><th>∞ (clamped)</th><th>1,00E+08</th><th>1,00E+06</th><th>1,00E+04</th><th>0 (hinged)</th></tr></thead><tbody><tr><td>1</td><td>3.49</td><td>3.49</td><td>3.46</td><td>3.42</td><td>3.41</td></tr><tr><td>2</td><td>11.21</td><td>11.21</td><td>11.08</td><td>10.99</td><td>10.99</td></tr><tr><td>3</td><td>24.81</td><td>25.05</td><td>25.03</td><td>25.01</td><td>25.00</td></tr><tr><td>4</td><td>27.60</td><td>27.59</td><td>27.44</td><td>27.22</td><td>27.21</td></tr><tr><td>5</td><td>46.62</td><td>46.60</td><td>45.93</td><td>44.38</td><td>44.31</td></tr><tr><td>6</td><td>80.35</td><td>80.32</td><td>78.72</td><td>74.23</td><td>73.99</td></tr><tr><td>7</td><td>106.15</td><td>106.07</td><td>97.50</td><td>79.85</td><td>79.60</td></tr><tr><td>8</td><td></td><td></td><td>104.72</td><td>85.96</td><td>85.11</td></tr><tr><td>9</td><td></td><td></td><td></td><td>87.06</td><td>87.14</td></tr><tr><td>10</td><td></td><td></td><td></td><td>93.89</td><td>93.64</td></tr><tr><td>11</td><td></td><td></td><td></td><td>95.57</td><td>95.29</td></tr><tr><td>12</td><td></td><td></td><td></td><td>105.82</td><td>105.50</td></tr></tbody></table>

*grey: vertical in-plane mode

<table><caption>**Table 3** Quality criteria $L_r$ comparing numerical and experimental mode shapes</caption><thead><tr><th rowspan="2">Experimental mode number</th><th colspan="5">k<sub>r</sub> [Nmm]</th></tr><tr><th>∞ (clamped)</th><th>1,00E+08</th><th>1,00E+06</th><th>1,00E+04</th><th>0 (hinged)</th></tr></thead><tbody><tr><td>1</td><td>0.20%</td><td>0.20%</td><td>0.25%</td><td>0.32%</td><td>0.32%</td></tr><tr><td>2</td><td>0.83%</td><td>0.83%</td><td>0.95%</td><td>1.07%</td><td>1.07%</td></tr><tr><td>3</td><td>1.50%</td><td>1.51%</td><td>1.72%</td><td>2.36%</td><td>2.39%</td></tr><tr><td>4</td><td>2.12%</td><td>2.13%</td><td>3.22%</td><td>5.01%</td><td>5.07%</td></tr><tr><td>5</td><td>5.23%</td><td>5.33%</td><td>14.19%</td><td>44.87%</td><td>45.36%</td></tr><tr><td>6</td><td>13.1%</td><td>13.53%</td><td></td><td></td><td></td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>**Figure 10** Deformed and original states of the structure for the 5th mode, obtained with the “hinged” numerical model: (a) 3d-view; (b) top view. The colours correspond to different quantities of out-of-plane displacements along z.</figcaption></figure>
# 6 Conclusion

This paper presented an investigation aiming to evaluate the out-of-plane rotational stiffness $k_r$ of riveted joints in a metallic lattice beam. An experimental modal analysis was conducted and the results were confronted with modal properties obtained from a finite element model. A first reference model with rigid joints was built, yielding a satisfactory agreement with the experiments. A parametric study was then conducted, taking only the joint stiffness $k_r$ at the web member ends as parameter. It was shown that decreasing $k_r$ resulted in a less good agreement between numerical and experimental results. This conclusion was based on two indicators. The first indicator is the quality criteria measuring the gap between numerical and experimental mode shapes. The second indicator is the presence or absence of additional numerical modes, dominated by the vibration of web members. These modes were not measured in the frequency range of interest. Their appearance in a numerical model suggests that this model is not valid.

This study offers two perspectives. First, if experimental modal analysis is applied to metallic riveted structures to get information on the stiffness of their joints, the eigenfrequencies are not the most relevant indicator. Careful analysis of the mode shapes is essential, and the deformations of the web members are the key parameters accounting for the gap between numerical and experimental results. Second, the single-riveted joints of lattice beams seem to be best-represented using clamped connections. This result would induce a decrease of the estimated buckling risk based on an eigenvalue analysis. Case studies still need to be carried out to evaluate the gain of stability when modelling the riveted joints as clamped instead of hinged. This could then effectively minimize strengthening measures linked with retrofitting operations.

# 7 Acknowledgments

This research is part of a PhD funded by AREP, subsidiary of SNCF, and the French Association for Research and Technology (ANRT).

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