1 Introduction

1.1 Background

The main aim of the Norwegian research project WOODSOL is to develop industrialised structural solutions based on moment-resisting timber frames and hollow box timber decks for urban multi-storey buildings allowing for greater architectural flexibility. The moment-resisting frames, the hollow box timber decks, the moment-resisting connections and the acoustic properties of the buildings are the research objectives of the project. The structural system consists of moment-resisting timber frames with continuous columns and hollow box timber decks connected to columns via semi-rigid beam-to-column connections.

The main purpose of a building's structural system is to carry vertical and lateral loads to the foundation. The type of structural system is chosen taking architectural and structural restrictions into account and a system based on moment-resisting timber frames with semi-rigid beam-to-column connections can be an interesting alternative. Timber frames offer many opportunities such as constructability, low environmental footprint and high strength-to-mass ratio (Gohlich et al. 2018; Kasal et al. 2014). Hollow box timber decks can be very effective floor elements and are further improved when connected by moment-resisting connections to the columns. Malo and Köhler (2013) showed that semi-rigid end restraints of beams can be exploited to achieve enhanced performance against human-induced vibrations and the possibility to increase spans between load bearing structures.

In timber engineering, the connections between structural elements are a crucial factor in the design of the structure. The overall strength and stiffness of the structures will typically be governed to a significant extent by the strength and stiffness of the connections. The most widely used connection types in timber engineering are laterally and axially loaded fasteners. Laterally loaded fasteners like dowel-type or bolted-type connections with slotted-in steel plates are commonly used as a connection between two or more structural elements. Dowel-type connections are designed to carry and withstand axial and shear forces. Such connections can be designed to carry bending moments. Solarino et al. (2017) performed experimental tests on typical beam-to-column moment-resisting dowel-type connection. The measured rotational stiffness in the range of 0.1 \({M}_{max}<M<\) 0.4 \({M}_{max}\) was 696 kNm/rad.

Lam et al. (2010) evaluated the performance of moment-resisting bolted timber connections with reinforcing wood screws inserted perpendicular to grain. The rotational stiffness of such a connection ranged from 850 to 2060 kNm/rad. Brandon and Leijten (2014) presented a study of two 3-member densified veneer wood reinforced timber connections with an inter-connecting steel plate used as middle member. Studies have shown that reinforcing dowel-type timber connections with ‘densified veneer wood’ and using expanded tube fasteners result in connections with superior structural properties compared to all conventional connections. The experimental tests of such a connection showed rotational stiffness of 1000 kNm/rad.

Axially loaded fasteners, like self-tapping screws (STS) or glued-in rods utilise the high axial stiffness of the fasteners. Komatsu et al. (2019) presented an experimental and analytical investigation of beam-to-column moment-resisting connection with STS and steel side plates. The beam-to-column moment-resisting connection showed initial stiffness of 1250 kNm/rad and an average moment resistance of 49.35 kNm. Leimcke et al. (2016) tested beam-to-column moment-resisting connection based on friction damping device and glued-in rods under cyclic loading. The test results showed high-energy dissipation without degradation of the timber beam or column. Andreolli et al. (2011) investigated the mechanical characteristics of moment-resisting connection with glued-in rods and steel end-plates suitable for different configurations of heavy timber frames. Two connection configurations were tested under monotonic and cyclic loading. The highest recorded rotational stiffness of the connection was approx. 2240 kNm/rad and 460 kNm/rad at failure. Buchanan et al. (2008) developed and showed the conceptual solution for a hybrid beam-column timber connection, based on the combination of post-tensioning and internal dissipaters (e.g. epoxied mild steel bars). The developed beam-to-column connection based on post-tensioning and internal dissipaters demonstrated high-seismic performance and large inelastic displacements (high-ductility demand).

To increase the span length of hollow box timber decks and to satisfy the structural requirements of buildings, the semi-rigid connection stiffness should be sufficient. The research presented in Vilguts et al. (2020) concluded that the connection stiffness required for multi-storey timber buildings with moment-resisting frames as the only lateral load resisting system should be around 10,000–15,000 kNm/rad. The moment-resisting timber connection with screwed-in, inclined threaded rods with wood screw threads and steel coupling parts has been presented in Lied and Nordal (2016) and Vilguts et al. (2018). Based on tests of these connection types, rotational stiffness values up to approx. 8000 kNm/rad can be achieved for two planes of rods. Therefore, the required stiffness values can be achieved with approx. 2–6 rod planes. Finally, moment-resisting timber frames may be used as a primary or secondary stabilising system in timber buildings in combinations with shear walls or truss systems depending on the design requirements, providing additional lateral stiffness to a building or increasing spans between load bearing constructions.

1.2 Outline

In the present paper, the experimental results from full-scale tests of a moment-resisting connection with inclined threaded rods and steel coupling parts are presented. The connections as shown in Fig. 1a were subjected to both cyclic and monotonic loading and the rotational stiffness, moment resistance and energy dissipation in the low-intensity domain were quantified. The experimental results from monotonic loading tests were compared to an analytical component model (Stamatopoulos and Malo 2020a). Moreover, a mock-up frame assembly with columns and hollow box timber decks connected with the moment-resisting connections, see Fig. 1b, was used to evaluate the properties of the connections in the frame assembly. The frame assembly was subjected to cyclic lateral loading and the lateral stiffness, energy dissipation and rotational stiffness of the connections were evaluated. In addition, the energy dissipation was estimated in the longitudinal and transversal directions of the frame assembly using the modal hammer technique followed by experimental modal analysis. The experimental results from the tests were input to finite element (FE) simulations, where the experimentally measured rotational stiffness and energy dissipation of the beam-to-column connections were used to model the behaviour of the connections in the mock-up frame assembly. Finally, the static and dynamic properties from experimental tests were evaluated and compared with results from FE modelling.

Fig. 1
figure 1

a Moment resisting connection with inclined threaded rods and steel coupling parts comprising of L-profiles inter-connected with friction bolts. b Mock-up frame assembly, photo: SINTEF/A.-L.Bakken

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2 Materials and methods

2.1 Moment-resisting connections

2.1.1 Experimental set-up and specimens

Three full-scale moment-resisting timber connections were tested under cyclic loading, to evaluate rotational stiffness and energy dissipation. One out of these three tests was tested until failure to explore the complete behaviour with respect to rotational stiffness and moment resistance. The experimental set-up is shown in Fig. 2 with corresponding pictures in Fig. 3. In total, 3 columns, 2 beams and 3 steel coupling parts were used to assemble the connection specimens.

Fig. 2
figure 2

Experimental set-up: a beam-to-column moment-resisting connection, b technical layout of the connection, c section A-A, d location of instrumentation

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Fig. 3
figure 3

Detailed experimental set-up: a experimental set-up, b steel coupling part, c load application, d bottom-view of the connection, e locations of LVDTs

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The beam-to-column moment-resisting connection consists of two timber beams, steel coupling parts and a timber column. The threaded rods at the beam-side were installed with rod-to-grain angle of 10°, while in the column-side 55 and 70° were used, as shown in Fig. 2b. The threaded rods were screwed in predrilled holes with diameter 17 mm. The rods were manufactured with two types of threads at their ends: wood screw threads in one end and M20 metric threads at the other end, as shown in Fig. 4. The inner and outer diameter of the wood screw threads were \({d}_{1}=\) 16.1 mm and \(d=\) 22 mm, respectively. The mean ultimate strength of the rods was \({f}_{u,mean}=\) 952 MPa (Lied and Nordal 2016).

Fig. 4
figure 4

Threaded rod

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The steel coupling parts at the beam-side and column-side were connected to the rods by use of purpose-made steel washers and M20 nuts, as shown in Fig. 2b. The steel coupling parts were made from standard L200 × 200 × 16 profiles (strength class S355) with oversized holes, to allow better fitting to the inclined threaded rods. The steel coupling parts were fastened to the column and beams with 5 mm and 20 mm gaps respectively, as shown in Fig. 2b. These intended gaps ensured that the forces were transferred solely by the threaded rods. The beam- and column-side steel coupling parts were connected with two M30 high strength friction bolts of strength class 12.9. The applied pre-stressing torque of the bolts was 2500 Nm. The rod-to-grain angles \({\alpha }_{i}\), embedment lengths of rods \({l}_{i}\), and the free length of the rods \({l}_{0,i}\) (i.e. the distance between the entrance point on wood and the fixing point in the steel plates) for tests are summarized in Table 1. The values in Table 1 are given for positive moment according to Fig. 2a.

Table 1 Parameters for experimental set-up according to Fig. 2
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Columns and beams were glued-laminated timber (glulam) elements made from Norway spruce (Picea abies) with strength class GL30c according to EN 14080:2013 (European Committee for Standardization 2013) and lamination thickness 45 mm. The cross-sectional dimensions of the columns were 405 × 450 mm2 (i.e. block glued glulam), while the beams were 140 × 405 mm2, confer Fig. 2a, c. The timber specimens were conditioned at temperature of 20 ºC and 65% relative humidity, resulting in approximately 12% moisture content in the wood.

The supports at both ends of the column were pinned. To prevent splitting perpendicular to grain, steel brackets were mounted at both supports, as shown in Fig. 3 (a). The point loading was applied on the beams by a clamp of two aluminium profiles (Fig. 3 (c)), which allowed both positive and negative point loading.

The deformation in beams, column and steel coupling parts was measured by a total of 12 displacement transducers (LVDTs), confer Fig. 2d. The relative horizontal deformation between the column and the beams was measured with four LVDTs (TD1, TD2, TD3, TD4). Four LVDTs measured the relative horizontal deformation between the column and the steel coupling parts at column-side (CD1, CD2, CD3, CD4), and four LVDTs measured the relative horizontal deformation between the beam and the steel coupler at beam-side (BD1, BD2, BD3, BD4). The total rotation of connection, column and beam was calculated accordingly:

$${\uptheta }_{y,t}=\frac{\left(\frac{{\delta }_{TD1}+{\delta }_{TD2}}{2}\right)-\left(\frac{{\delta }_{TD3}+{\delta }_{TD4}}{2}\right)}{{z}_{t}}$$
(1)
$${\uptheta }_{y,c}=\frac{\left(\frac{{\delta }_{CD1}+{\delta }_{CD2}}{2}\right)-\left(\frac{{\delta }_{CD3}+{\delta }_{CD4}}{2}\right)}{{z}_{c}}$$
(2)
$${\uptheta }_{y,b}=\frac{\left(\frac{{\delta }_{BD1}+{\delta }_{BD2}}{2}\right)-\left(\frac{{\delta }_{BD3}+{\delta }_{BD4}}{2}\right)}{{z}_{b}}$$
(3)

where \({z}_{t}, {z}_{c}, {z}_{b}-\) are the lever arms between LVDTs for the total rotation of connection, column and beam, see Fig. 2 and \({\delta }_{i}-\) are the measured relative displacements.

2.1.2 Loading protocol

In total, four loading protocols were applied, as shown in Table 2:

  • Fully reversed cyclic loading in the range of + 30 to – 30 kNm. The cyclic rotational stiffness \({K}_{\theta ,cyc}\) was obtained by fitting a straight line to the hysteresis loops giving a representative stiffness for a connection subjected to cyclic alternate loading (typical in structures subjected to vibration). The area enclosed in each hysteresis loop is the energy dissipation per cycle \({E}_{d}\). The load was applied with a quasi-static rate with a frequency of \(f=\) 0.008 Hz. The range of the applied moment (± 30kNm) was considered as a representative service load and approximated as 40% of the estimated moment resistance according to EN 26891:1991 (European Committee for Standardization 1991).

  • Cyclic loading with positive moment ranging from + 7.5 to + 30 kNm and cyclic loading with negative moment ranging from – 7.5 to – 30 kNm. These loading schemes do not give alternating signs, that is, the moment does not go through zero moment. The cyclic rotational stiffness and the energy dissipation are determined for each side separately. The load was applied with a quasi-static rate with a frequency of \(f=\) 0.015 Hz.

  • Monotonic loading until failure according to EN 26891:1991 (European Committee for Standardization 1991).

Table 2 Imposed moment vs rotation for the moment-resisting beam-to-column connections according to the various loading schemes
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All three columns with one double beam were tested according to the testing protocol in the following order: cyclic loading with positive moment, cyclic loading with negative moment and fully reversed cyclic loading. Exception was the third column, where after the cyclic testing protocol described above, the specimen was subjected to monotonic loading until failure.

2.1.3 Equivalent viscous damping

To quantify energy dissipation in the connection subjected to cyclic loading, the equivalent viscous damping ratio is estimated according to Eq. (4) (Chopra 2012):

$${\upxi }_{eq}=\frac{1}{4\pi }\frac{{E}_{d}}{\Delta {E}_{el}}$$
(4)

where \({E}_{d}\) is the dissipated energy per cycle and \(\Delta {E}_{el}-\) is the corresponding change in elastic energy per cycle. The dissipated energy \({E}_{d}\) per cycle can be found from the measured enclosed area in the hysteresis loop, visualised in Table 2. The maximum elastic change \(\Delta {E}_{el}\) during a cycle is:

  • In case of moment-rotation:

    $$\Delta {E}_{el}=\frac{1}{2}\frac{{M}_{a}^{2}}{{K}_{\uptheta ,\mathrm{cyc}}}$$
    (5)

  • In case of force–displacement:

    $$\Delta {E}_{el}=\frac{1}{2}\frac{{F}_{a}^{2}}{{K}_{ax,\mathrm{cyc}}}$$
    (6)

    where \({M}_{a}=1/2\times \left({M}_{max}-{M}_{min}\right)\) and \({F}_{a}=1/2\times \left({F}_{max}-{F}_{min}\right)\), \({M}_{max}\) and \({M}_{min}-\) are the maximum and minimum moments in the hysteresis loop, \({F}_{max}\) and \({F}_{min}-\) are the maximum and minimum applied forces in the hysteresis loop, \({K}_{\uptheta ,\mathrm{cyc}},{K}_{ax,cyc}-\) are the linear stiffnesses obtained by fitting a straight line to all points in the hysteresis loop, using the method of least squares.

2.1.4 Analytical methods

2.1.4.1 Rotational stiffness of connection

The analytical model proposed in Stamatopoulos and Malo (2020a) is used in the present paper to calculate the rotational stiffness and resistance of the connection. The analytical model considers the connection in three separate parts: column-side, beam-side and steel coupling part.

The horizontal force component \({F}_{x}\) is obtained by moment equilibrium, as shown in Fig. 2:

$${F}_{x}=\frac{M}{z}$$
(7)

where \(M-\) is the moment acting on the connection and \(z-\) is the lever arm of the connection.

The rotational stiffness about y-axis of the connection at the column-side is obtained using the following equation (Stamatopoulos and Malo 2020a):

$${K}_{\uptheta ,\mathrm{y},\mathrm{c}}=\frac{{M}_{y}}{{\uptheta }_{y,c}}=\frac{{z}_{c}^{2}}{\left({S}_{xx,c}^{(c1-c2)}+{S}_{xx,c}^{(c3-c4)}\right)+\left({S}_{xz,c}^{(c3-c4)}-{S}_{xz,c}^{(c1-c2)}\right)\times \frac{{z}_{c}}{2\times {L}_{v}}}$$
(8)
$${S}_{xx,c}^{(c1-c2)}=\frac{{\mathrm{cos}}^{2}{\alpha }_{c1}/{K}_{ax,c2}+{\mathrm{cos}}^{2}{\alpha }_{c2}/{K}_{ax,c1}}{{\left(\mathrm{cos}{\alpha }_{c1}\times \mathrm{sin}{\alpha }_{c2}+\mathrm{cos}{\alpha }_{c2}\times \mathrm{sin}{\alpha }_{c1}\right)}^{2}}$$
(9)
$${S}_{xx,c}^{(c3-c4)}=\frac{{\mathrm{cos}}^{2}{\alpha }_{c3}/{K}_{ax,c4}+{\mathrm{cos}}^{2}{\alpha }_{c4}/{K}_{ax,c3}}{{\left(\mathrm{cos}{\alpha }_{c3}\times \mathrm{sin}{\alpha }_{c4}+\mathrm{cos}{\alpha }_{c4}\times \mathrm{sin}{\alpha }_{c3}\right)}^{2}}$$
(10)
$${S}_{xz,c}^{(c1-c2)}=\frac{\mathrm{cos}{\alpha }_{c1}\times \mathrm{sin}{\alpha }_{c1}/{K}_{ax,c2}-\mathrm{cos}{\alpha }_{c2}\times \mathrm{sin}{\alpha }_{c2}/{K}_{ax,c1}}{{\left(\mathrm{cos}{\alpha }_{c1}\times \mathrm{sin}{\alpha }_{c2}+\mathrm{cos}{\alpha }_{c2}\times \mathrm{sin}{\alpha }_{1}\right)}^{2}}$$
(11)
$${S}_{xz,c}^{(c3-c4)}=\frac{\mathrm{cos}{\alpha }_{c3}\times \mathrm{sin}{\alpha }_{c3}/{K}_{ax,c4}-\mathrm{cos}{\alpha }_{c4}\times \mathrm{sin}{\alpha }_{c4}/{K}_{ax,c3}}{{\left(\mathrm{cos}{\alpha }_{c3}\times \mathrm{sin}{\alpha }_{c4}+\mathrm{cos}{\alpha }_{c4}\times \mathrm{sin}{\alpha }_{c3}\right)}^{2}}$$
(12)

where \({K}_{ax,c1}, {K}_{ax,c2}, {K}_{ax,c3}, {K}_{ax,c4}\) \({K}_{ax,j}\) are the axial stiffness of rods c1-c4 (see Sect. 2.1.4.3), \({z}_{c}-\) is the lever arm between threaded rods at column-side, \({L}_{v}=M/F-\) (see Fig. 2a) is denoted as the shear length and \(\mathrm{cos}\alpha -\) and \(\mathrm{sin}\alpha -\) are the angle between the grain and threaded rods (see Fig. 2b).

The rotational stiffness about y-axis of the connection at the beam-side is given by the following equation (Stamatopoulos and Malo 2020a):

$${K}_{\uptheta ,\mathrm{y},\mathrm{b}}=\frac{{M}_{y}}{{\uptheta }_{y,b}}=\frac{{z}_{b}^{2}}{\left({S}_{xx,b1}+{S}_{xx,b2}\right)+\left({S}_{xz,b2}-{S}_{xz,b1}\right)\times \frac{{z}_{b}}{2\times {L}_{v}}}$$
(13)
$${S}_{xx,b1}=\frac{{\mathrm{sin}{\alpha }_{b1}}^{2}}{{K}_{v,b1}}+\frac{{\mathrm{cos}{\alpha }_{b1}}^{2}}{{K}_{ax,b1}}$$
(14)
$${S}_{xx,b2}=\frac{{\mathrm{sin}{\alpha }_{b2}}^{2}}{{K}_{v,b2}}+\frac{{\mathrm{cos}{\alpha }_{b2}}^{2}}{{K}_{ax,b2}}$$
(15)
$${S}_{xz,b1}=\mathrm{sin}{\alpha }_{b1}\times \mathrm{cos}{\alpha }_{b1}\times \left(\frac{1}{{K}_{v,b1}}-\frac{1}{{K}_{ax,b1}}\right)$$
(16)
$${S}_{xz,b2}=\mathrm{sin}{\alpha }_{b2}\times \mathrm{cos}{\alpha }_{b2}\times \left(\frac{1}{{K}_{ax,b2}}-\frac{1}{{K}_{v,b2}}\right)$$
(17)

where \({K}_{ax,b1},{K}_{ax,b2}, {K}_{v,b1}, {K}_{v,b2}\) are the axial and lateral stiffness of threaded rods (see Sect. 2.1.4.3), \({z}_{b}-\) is the lever arm between threaded rods at beam-side.

The steel connector has coupling parts both at the column- and beam-side, which are inter-connected by friction bolts. The rotational stiffness of the steel connector can be measured or derived from FE model, see Appendix A in Electronic Supplementary Information.

The total rotational stiffness of the connection can be calculated according to Eq. (18):

$${K}_{\uptheta ,\mathrm{y}}={\left(\frac{1}{{K}_{\uptheta ,\mathrm{c}}}+\frac{1}{{K}_{\uptheta ,\mathrm{b}}}+\frac{1}{{K}_{\uptheta ,\mathrm{con}}}\right)}^{-1}$$
(18)
2.1.4.2 Capacity of threaded rods and column

The threaded rods are mainly axially loaded and the axial capacity per threaded rod is given by the following equation:

$${F}_{ax,R}=\frac{{n}_{ef}}{n}\times \underset{}{\mathrm{min}}\left\{\begin{array}{c}{F}_{ax,a,R}\\ {F}_{tens,R}\end{array}\right.$$
(19)

where \({n}_{ef}={n}^{0.9}-\) is the effective number of threaded rods acting together according to EN1995-1–1 (European Committee for Standardization 2010) and \({F}_{tens,R}={A}_{s}\times {f}_{u,mean}\) is the tensile capacity of each rod.

On the column-side, the axial forces in each rod can be estimated according to Eqs. (20–23) (Stamatopoulos and Malo 2020a):

$${F}_{ax,c1}=\frac{1}{n}\times \frac{\mathrm{cos}{\alpha }_{c2}+\mathrm{sin}{\alpha }_{c2}\times \frac{{z}_{c}}{2\times {L}_{v}}}{\mathrm{cos}{\alpha }_{c1}\times \mathrm{sin}{\alpha }_{c2}+\mathrm{cos}{\alpha }_{c2}\times \mathrm{sin}{\alpha }_{c1}}\times \frac{M}{{z}_{c}}$$
(20)
$${F}_{ax,c2}=\frac{1}{n}\times \frac{\mathrm{cos}{\alpha }_{c1}-\mathrm{sin}{\alpha }_{c1}\times \frac{{z}_{c}}{2\times {L}_{v}}}{\mathrm{cos}{\alpha }_{c1}\times \mathrm{sin}{\alpha }_{c2}+\mathrm{cos}{\alpha }_{c2}\times \mathrm{sin}{\alpha }_{c1}}\times \frac{M}{{z}_{c}}$$
(21)
$${F}_{ax,c3}=-\frac{1}{n}\times \frac{\mathrm{cos}{\alpha }_{c4}-\mathrm{sin}{\alpha }_{c4}\times \frac{{z}_{c}}{2\times {L}_{v}}}{\mathrm{cos}{\alpha }_{c3}\times \mathrm{sin}{\alpha }_{c4}+\mathrm{cos}{\alpha }_{c4}\times \mathrm{sin}{\alpha }_{c3}}\times \frac{M}{{z}_{c}}$$
(22)
$${F}_{ax,c4}=-\frac{1}{n}\times \frac{\mathrm{cos}{\alpha }_{c3}+\mathrm{sin}{\alpha }_{c3}\times \frac{{z}_{c}}{2\times {L}_{v}}}{\mathrm{cos}{\alpha }_{c3}\times \mathrm{sin}{\alpha }_{c4}+\mathrm{cos}{\alpha }_{c4}\times \mathrm{sin}{\alpha }_{c3}}\times \frac{M}{{z}_{c}}$$
(23)

On the beam-side, the axial forces in each rod can be estimated according to Eqs. (24) and (25) (Stamatopoulos and Malo 2020a):

$${F}_{ax,b1}=\frac{1}{n}\times \left(cos{\alpha }_{b1}+sin{\alpha }_{b1}\times \frac{z}{2\times {L}_{v}}\right)\times \frac{M}{z}$$
(24)
$${F}_{ax,b2}=-\frac{1}{n}\times \left(cos{\alpha }_{b2}+sin{\alpha }_{b2}\times \frac{z}{2\times {L}_{v}}\right)\times \frac{M}{z}$$
(25)
2.1.4.3 Analytical predictions of axial and lateral stiffness and capacity of screwed-in threaded rods

The axial stiffness of threaded rod is one of the most important parameters. The withdrawal stiffness of threaded rods can be approximated by Eqs. (26) and (27) (Stamatopoulos and Malo 2020b):

$${K}_{ser,ax}\approx \frac{50000\times {\left(\frac{d}{20}\right)}^{2}\times {\left(\frac{{\rho }_{m}}{470}\right)}^{2}\times {k}_{lenght,K}}{0.40\times \mathrm{cos}{}^{2.3}\alpha +\mathrm{sin}{}^{2.3}\alpha }$$
(26)
$${k}_{lenght,K}=\underset{}{\mathrm{min}}\left[{\left(\frac{l}{300}\right)}^{0.75},1.0\right]$$
(27)

where \(d-\) is the outer-thread diameter of the rod in mm, \({\rho }_{m}-\) is the wood density in kg/m3, \(\alpha -\) is the rod-to-grain angle and \(l-\) is the embedment length in mm.

The axial stiffness of the free part (non-embedded) of the threaded rod is given by the following equation:

$${K}_{ax,l0}={A}_{s}\times {E}_{s}/{l}_{0}$$
(28)

where \({A}_{s}=\pi \times {d}_{1}^{2}/4\); \({E}_{s}=\) 210,000 MPa, \({d}_{1}-\) is the inner-diameter of threaded rod, \({l}_{0}-\) is the free length of the rod not embedded in timber (i.e. the length between the entrance point in wood and the fastening point in the steel coupling parts).

The total axial stiffness of the threaded rods is given by the following equation:

$${K}_{ser,ax,tot}=\frac{{K}_{ser,ax}\times {K}_{ax,l0}}{{K}_{ser,ax}+{K}_{ax,l0}}$$
(29)

The lateral stiffness of a threaded rod subjected to lateral loading depends on the rotation of the end of the rod at the connecting point to the steel coupling part (Stamatopoulos and Malo 2020a). For fixed end, the following expression can be used:

$${K}_{ser,v,tot}=\frac{3\times m\times {k}_{v}\times {l}_{ch}\times \left({\uplambda }_{0}+m\right)}{{\uplambda }_{0}^{4}+4\times {\uplambda }_{0}^{3}\times m+6\times {\uplambda }_{0}^{2}\times m+6\times {\uplambda }_{0}\times m+3\times {m}^{2}}$$
(30)

where \({k}_{v}\approx\) 300 N/mm2 is the foundation modulus of timber (Qazi 2020), \(m={d}_{net}^{4}/{d}_{1}^{4}\) \({\uplambda }_{0}={l}_{0}/{l}_{c}\), \({l}_{ch}=\sqrt[4]{4\times {E}_{s}\times {I}_{s}/{k}_{v}}\). The flexural stiffness of the embedded part screw is \({E}_{s}\times {I}_{s}={E}_{s}\times \pi \times {d}_{1}^{4}/64\), \({d}_{net}=\) 0.9 \(\times\) 20 = 18 mm is the diameter of the non-embedded part.

A simplified expression, derived in (Stamatopoulos and Malo 2020b), can be used to estimate mean withdrawal capacity of threaded rods:

$${F}_{ax,a,R}\approx 15.0\times d\times l\times \left(\frac{{\rho }_{m}}{470}\right)$$
(31)

2.2 Mock-up frame assembly

2.2.1 Experimental set-up and specimens

The mock-up frame assembly shown in Fig. 5 was used to evaluate the properties of connections in moment-resisting frame. The mock-up frame assembly consisted of six glulam columns and two hollow 4.7 m long box timber deck elements. The connections were similar to those presented in Sect. 2.1, as shown in Fig. 5b, c. A cross-section of two parallel floor elements in direction of the span is shown in Fig. 5d. Each of the deck elements consisted of two external parallel glulam beams (1) of GL30c with cross section of 140 × 405 mm2 and three internal glulam beams (2) of GL28c with cross section of 66 × 405 mm2. The top (3) and bottom (4) flanges were made from Kerto-Q LVL plates with thickness of 43 mm and 61 mm, respectively. The elastic material properties are given in Table 3.

Fig. 5
figure 5

Mock-up frame assembly set-up. a Overview of the construction, b connection between floors and centre columns (C2), c connection between floors and edge columns (C1 and C3), d cross-section of the floor elements. Photo a–c: SINTEF/A.-L.Bakken

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Table 3 Material properties (Kristoffersen and Bjørge 2017)
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The flange plates (Kerto-Q) were glued to beams with phenol-resorcinol adhesive and therefore rigid behaviour at the interface was assumed. The deck elements were installed to the columns with the bottom flange 2 m above the floor level.

The columns were 5.2 m high with a block-glued cross-section of 405 × 450 mm2. The columns used in this set-up were the same columns as used for the beam-to-column moment-resisting tests described in Sect. 2.1. The columns were installed on the concrete floor with brackets, which allowed rotation about y-axis, as shown in Fig. 5a, that is, the connections to the floor were pinned about the y-axis.

The hollow box decks elements were connected to the columns with the moment-resisting connections described in Sect. 2.1. Two M30 high strength friction bolts (grade 12.9) with applied prestressing torque on the bolts of 2500 Nm were used. Both floor elements shared the central column (C2) with the double-sided connection shown in Fig. 5b, while the connections to the corner columns were a single-sided version of the same connection, as shown in Fig. 5c.

2.2.2 Loading protocol and instrumentation

To evaluate lateral stiffness, energy dissipation and some dynamic properties of the frame assembly, the following tests were performed:

  1. 1.

    Cyclic loading (loading–unloading) from 0 to + 11/12 kN. The load was applied two meters above the concrete floor. This loading was applied on each one of the columns (C1, C2, C3) separately, that is, three test series were performed in total. The load was applied with a quasi-static rate with a frequency of \(f=\) 0.02 Hz.

  2. 2.

    Experimental modal analysis (EMA), by use of the roving hammer technique using one reference accelerometer and a defined grid of hammering points. The impact loads were imposed at the level of the moment-resisting connections as shown in Fig. 6b; in the direction of moment-resisting connections (x-axis), and in the transversal direction (y-axis). An impact hammer with a soft rubber cap was used to excite the structure. Two impacts at each point of the grid were executed, recorded and averaged. The experimental modal analysis was limited to the two horizontal directions.

Fig. 6
figure 6

a 3D FE model of frame structure, b horizontal cross-section of the frame assembly showing the locations of accelerometers and excitation points

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The lateral deformations of the prototype frame construction were measured by a total of 9 LVDTs. Three LVDTs measured the absolute displacements (x-axis) of the columns (C1, C2, C3) two meters from the concrete floor (AD1, AD2, AD3). Two LVDTs for each column as illustrated in Fig. 5c for column C1 with RD1 and RD2 were used to measure the relative displacements between the columns and floor elements at the top and bottom levels of the moment-resisting connections, which allowed to determine their rotations. Additionally, for column C2, relative displacements were measured by LVDTs RD3 and RD4 and for column C3, relative displacements were measured by LVDTs RD5 and RD6.

2.2.3 Finite element analysis

Finite Element analysis of frame assembly was carried out by SAP2000 Finite Element software (SAP 2000; Wilson and Habibullah 1997), where the rotational stiffness of moment-resisting connections on the mock-up frame assembly is the emphasized issue. The obtained results are compared to the experimental tests. The layout of the 3D frame structure is shown in Fig. 6a.

The glulam columns were modelled with linear elastic beam elements with mean elastic moduli equal to: \({E}_{0,mean,c}=\) 13,000 MPa and \({G}_{mean,c}=\) 650 MPa for GL30c according to EN14080:2013 (European Committee for Standardization 2013), see Table 3. The floor elements were modelled as thick shell elements with effective bending stiffness of \(E{I}_{f,longitudinal}=\) 6.7 \(\times\) 1013 N \(\times\) mm2 in the longitudinal x-direction (Conta and Homb 2020) and \(E{I}_{f,transversal}=\) 4 \(\times\) 1013 N \(\times\) mm2 in the transversal y-direction (Kristoffersen and Bjørge 2017). These effective bending stiffness values in longitudinal and transversal directions were measured experimentally.

The moment-resisting connections between columns and floor elements were modelled as semi-rigid with respect to the rotational degrees of freedom (DOF) about x- and y-axis. Linear-elastic rotational springs with spring constants \({K}_{\uptheta ,x}\) and \({K}_{\uptheta ,y}\) represented the moment-resisting connections. The translational DOFs were fixed between columns and decks in all directions, while the rotation about z-axis was released (pinned). The rotational stiffness values \({K}_{\Theta ,\mathrm{y}}\) used for FE analysis were taken from the experimental tests of the moment-resisting connections. For the centric columns (C2), the rotational stiffness was \({K}_{\uptheta ,\mathrm{y}}\), but for the connections at the corner columns (C1 and C3) \({K}_{\uptheta ,\mathrm{y}}/2\) was used. The connections for the C2 columns consist of the double amount of threaded rods and steel coupling parts compared to the connections at the corner columns.

2.2.4 Analytical predictions of connection stiffness about x-axis

The rotational stiffness about x-axis \({K}_{\uptheta ,x}\) was estimated from the following expressions as the total connection rotational stiffness, column-side stiffness and beam-side stiffness, respectively:

$${K}_{\uptheta ,\mathrm{x}}=\frac{{K}_{\uptheta ,\mathrm{x},\mathrm{c}}\times {K}_{\uptheta ,\mathrm{x},\mathrm{b}}}{{K}_{\uptheta ,\mathrm{x},\mathrm{c}}+{K}_{\uptheta ,\mathrm{x},\mathrm{b}}}$$
(32)
$${K}_{\uptheta ,\mathrm{x},\mathrm{c}}=\frac{{M}_{x}}{{\uptheta }_{x,c}}={z}_{x,c}^{2}\times {\left(\frac{1}{{K}_{v,y,c}}+\frac{1}{{K}_{v,y,c}}\right)}^{-1}$$
(33)
$${K}_{\uptheta ,\mathrm{x},\mathrm{b}}=\frac{{M}_{x}}{{\uptheta }_{x,b}}={z}_{x,b}^{2}\times {\left(\frac{1}{{K}_{v,y,b}}+\frac{1}{{K}_{v,y,b}}\right)}^{-1}$$
(34)

where \({K}_{v,y,c}, {K}_{v,y,b}-\) are the lateral stiffness of threaded rods (see Sect. 2.1.4.3) and \({z}_{x,c},{z}_{x,b}-\) are the lever arms between threaded rods in column and beam, respectively.

2.2.5 Free damped vibrations of mock-up frame assembly

One major objective in the present research was to explore possible prediction of structural damping based on knowledge of mass, stiffness and damping characteristics of each of the components constituting a complete structure. The mock-up frame assembly, see Fig. 1b, was modelled with the FE model shown in Fig. 6a. The following values (in parenthesis) were used for the material damping in the various components; timber columns (0.5%) (Labonnote et al. 2013), hollow box timber decks (1.5%) (Conta and Homb 2020) and the stiffness and mass were modelled by their representative tabulated mean values. The moment-resisting connections were modelled as linear-elastic rotational springs with the average values of spring stiffness from the cyclic tests of the connections. The equivalent viscous damping of the connections in the FE model was modelled by the rotational damping coefficient (\(c\)) as stiffness-proportional damping (Chopra 2012):

$$c=\beta \times {K}_{\uptheta ,\mathrm{y}}$$
(35)

where \(\upbeta =2{\upxi }_{eq}/{\omega }_{n,x}\),\({\upxi }_{eq}-\) is the experimentally measured equivalent viscous damping for the connections (equivalent viscous damping assumed equal for all the connection in FE model), and \({\omega }_{n,x}=2\pi \times {f}_{n,x}\), \({f}_{n,x}-\) is the undamped fundamental eigenfrequency obtained from FE simulations along x-direction. In calculations of damping coefficient (\(c\)), the rotational stiffness of connections on centric columns (C2) \({K}_{\uptheta ,\mathrm{y}}\) was taken from cyclic experimental tests of the moment-resisting connections, while the rotational stiffness on the corner columns (C1 and C3) was taken as \({K}_{\uptheta ,\mathrm{y}}/2\).

Using the FE model and initial conditions (applied displacement on columns along x-axis), a damped free vibration was obtained in the x-direction of the mock-up frame assembly. The structural damping from the damped simulation was estimated using the method with logarithmic decrement, leading to Eq. (36) (Chopra 2012):

$$\upxi =\frac{1}{2\pi j}\mathrm{ln}\frac{{u}_{i}}{{u}_{i+j}}$$
(36)

where \(j-\) is the number of cycles for evaluation, \({u}_{i}-\) is the maximum displacement of \(i\)-th cycle and \({u}_{i+j}-\) is the (decreased) displacement of (\(i+j\))-th cycle.

From the free vibrational response of the FE model by applying the material and connection mechanical properties, the vibrational frequency and the structural damping for the structure were determined.

3 Results and discussion

3.1 Moment-resisting connection

3.1.1 Cyclic loading tests

Initially, cyclic tests were performed on the moment-resisting connections using the loading protocols presented in Sect. 2.1.2. For each cyclic loading case, 10 cycles were performed. The experimentally recorded moment-rotation hysteresis loops for each specimen are shown in Fig. 7, where the first column gives the total rotational stiffness for the test set-ups, while the second and third columns visualise the contributions from beam-side and column-side, respectively. Table 4 summarizes the experimentally recorded cyclic stiffness values, the (average) energy dissipation per cycle and the equivalent viscous damping ratios according to Eq. (5) for the connection and for each part separately.

Fig. 7
figure 7

Moment-rotation curves from cyclic tests of the connections

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Table 4 Experimental results of connections subjected to cyclic testing
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The mean rotational stiffness measured for fully reversed cyclic loading \({\mathrm{K}}_{\uptheta ,\mathrm{cyc}}=\) 4175 kNm/rad is slightly lower compared to mean values of the rotational stiffness for positive moment (\({\mathrm{K}}_{\uptheta ,+}=\) 4900 kNm/rad) and negative moment (\({\mathrm{K}}_{\uptheta ,-}=\) 4480 kNm/rad). Furthermore, the measured stiffness values from monotonic loading (4761 kNm/rad) are quite similar to all the values recorded from the cyclic tests. The mean rotational stiffness on the column-side connection ranges from 8341 to 8815 kNm/rad and on the beam-side connection, the mean rotational stiffness ranges from 34,367 to 41,939 kNm/rad.

The mean equivalent damping ratios are similar for fully reversed loading (4.2%), positive moment (4.7%) and negative moment (4.9%) with variability coefficient of 13.4%, 18.9% and 11.4%, respectively. The mean equivalent viscous damping ratios for the beam-side connection are approximately two times higher compared to column-side, both for fully reversed loading (5.7% vs 2.6%), cyclic loading under positive moment (6.3% vs 3.8%) and cyclic loading under negative moment (6.6% vs 2.4%). This finding may be related to the fact that more threaded rods are used on the beam-side possibly resulting in larger energy dissipation.

3.2 Mock-up frame assembly

3.2.1 Cyclic loading tests

The experimental results from cyclic loading tests are given in Fig. 8 in terms of force–displacement hysteresis loops for each loading point and displacement measurement. The displacements in x-direction were monitored by the LVDTs attached to the columns C1, C2 and C3, as shown in Fig. 5a. Consequently, Fig. 8 can be interpreted as a graphical 3 × 3 stiffness matrix. As it may be observed, the off-diagonal elements are fairly symmetric and indicate the degree of coupling effects between the three columns. The corresponding values for lateral stiffness (\({K}_{frame}\)), energy dissipation (\({E}_{d}\)), viscous damping coefficient (\({\xi }_{eq}\)), bending moments (\({M}_{c}\)) and rotational stiffness of connections (\({K}_{\uptheta ,\mathrm{y}}\)) are summarized in Table 5. Furthermore, the experimental results were compared with the FE model shown in Fig. 6a.

Fig. 8
figure 8

Force–displacement curves from lateral cyclic tests of the mock-up frame assembly

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Table 5 Experimental and FE results of mock-up frame assembly subjected to lateral cyclic loading
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As shown in Table 5, the lateral stiffness of the mock-up frame assembly for loading in C1, C2, C3 were measured to 2.43, 4.49, 2.33 kN/mm, respectively. The lateral stiffness is smallest when the load is applied on the corner columns, presumably due to torsion caused by the eccentric loading. Furthermore, the frame assembly showed considerable energy dissipation with equivalent viscous damping coefficients for all loading cases (3.11%, 2.62% and 4.65%).

3.2.2 Finite element evaluations

To examine the effects of the transversal (x-axis) rotational stiffness of the connections, a series of FE simulations of the frame assembly was carried out varying the ratio between the rotational stiffness about x-axis (transversal) and y-axis \({\mathrm{K}}_{\uptheta ,\mathrm{x}}/{\mathrm{K}}_{\uptheta ,\mathrm{y}}\). The rotational stiffness (\({\mathrm{K}}_{\uptheta ,\mathrm{y}}\)) about y-axis was fixed to 4200 and 2100 kNm/rad for centric and corner columns, respectively (i.e., the average rotational stiffness from the cyclic loading tests). The FE results are presented in Fig. 9 with corresponding lateral loading \(({\mathrm{F}}_{\mathrm{max}}\)) on every column according to Table 5. The obtained displacements from FE simulations for all three columns were normalized with displacements of middle (C2) column. The stiffness ratio range of \({\mathrm{K}}_{\uptheta ,\mathrm{x}}/{\mathrm{K}}_{\uptheta ,\mathrm{y}}=\) 0.2–0.3, which is highlighted in Fig. 9, gives good agreement regardless of load condition and is valid both for static and dynamic response. The rotational stiffness of connection about x-axis does not affect horizontal displacements when the loading is applied on the centric column, confer Fig. 9. Furthermore, the mean rotational stiffness of connection measured in the laboratory was 4175 kNm/rad, but for the mock-up frame assembly the measurements gave rotational stiffness from 4166 to 4535 kNm/rad. The mean rotational stiffness of connection for corner columns was measured to 1878 kNm/rad. The FE model with the stiffness ratio \({\mathrm{K}}_{\uptheta ,\mathrm{x}}/{\mathrm{K}}_{\uptheta ,\mathrm{y}}\) in the range 0.20–0.30 also gave good agreement for the fundamental eigenfrequency in the y-direction compared to the experimentally obtained value of 1.81 Hz (Table 6), see the rightmost plot in Fig. 9. The corresponding vibrational modes from FE simulations are shown in Fig. 10.

Fig. 9
figure 9

Lateral displacements according to FE analyses of the frame assembly as a function of the ratio \({\mathrm{K}}_{\uptheta ,\mathrm{x}}/{\mathrm{K}}_{\uptheta ,\mathrm{y}}\)

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Table 6 Fundamental eigenfrequencies and structural damping from EMA and FE simulations
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Fig. 10
figure 10

Mock-up frame assembly mode shapes from FE software SAP2000: a First mode in transversal y-direction with frequency of 1.80 Hz; b Second mode in longitudinal x-direction with frequency of 3.34 Hz

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3.2.3 Free damped vibration evaluation

The fundamental eigenfrequencies and damping ratios with respect to translation in x- and y-directions obtained from experimental modal analysis and FE simulation are given in Table 6. The fundamental eigenfrequency with EMA was recorded to be 1.81 Hz along the y-direction and 3.55 Hz along the x-direction with structural damping ratios of 2.1% and 3.9%, respectively. No experiment on energy dissipation in the connections for transverse direction (y-direction) was performed, and hence no comparison of FE and EMA is available for y-direction.

To evaluate the structural damping with FE simulations as discussed in Sect. 2.2.5, the material damping values of the timber columns and hollow box timber decks were applied. Additionally, the damping coefficients of the moment-resisting connections were estimated according to Eq. (35) and all values are presented in Table 7. The free damped vibration results from FE simulation are presented in Fig. 11 in terms of displacement vs time and compared with free damped vibration test results from values obtained with EMA by applying the equation of motion for damped structures:

$$u\left(t\right)={e}^{-{\xi }_{x}{\omega }_{n,x}t}\left[u\left(0\right)\mathrm{cos}{\omega }_{d,x}t+\frac{\dot{u}\left(0\right)+{\xi }_{x}{\omega }_{n,x}u\left(0\right)}{{\omega }_{d,x}}\mathrm{sin}{\omega }_{d,x}t\right]$$
(37)
$${\omega }_{d,x}={\omega }_{n,x}\sqrt{1-{{\xi }_{x}}^{2}}$$
(38)

where \({\xi }_{x}-\) is the structural damping from EMA, \({\omega }_{n,x}-\) is the natural frequency, \(u\left(0\right)-\) is the initial displacement, \({\omega }_{d,x}-\) is the damped natural frequency, \(\dot{u}\left(0\right)-\) is the initial velocity.

Table 7 Material and equivalent viscous damping ratios for FE simulation
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Fig. 11
figure 11

EMA and FE free damped vibration results in x-direction

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The initial displacement on the centric and corner columns at the level of hollow box timber decks \(u\left(0\right)=\) 5.0 mm and initial velocity \(\dot{u}\left(0\right)=\) 0 m/s were applied to measure free damped vibrations. The structural damping of FE frame assembly model according to Eq. (36) and Fig. 11 is 3.88%.

As can be seen in Table 6, as well as Fig. 11, a minor difference in the fundamental frequency \({f}_{n,x}\) between the FE model (with measured component values) and the EMA results (of the mock-up frame assembly) is present. The FE model gives \({f}_{n,x}=\) 3.34 Hz, while \({f}_{n,x}=\) 3.55 Hz is obtained from EMA. This is probably due to small differences in the rotational stiffness of the connections. The resulting deviation in frequency is consistent with the fact that the connection stiffness obtained from the mock-up (and consequently in the EMA) were slightly larger than the stiffness measured in the connection component tests. It should also be kept in mind that there are experimental variations in the performance of the connections, both in the component tests as well as in the mock-up frame assembly tests. The use of material damping and equivalent viscous damping from the cyclic connection tests show very good agreement, and this approach can successfully be used to predict the structural damping of complete frame assemblies.

4 Conclusion

The performance of beam-to-column moment-resisting timber connections with inclined threaded rods and steel coupling parts connected with friction bolts has been investigated by use of full-scale tests and a mock-up frame assembly. The moment resisting connections were subjected both to cyclic and monotonic loading. The rotational stiffness and the energy dissipation properties were determined on the basis of three full-scale cyclic tests. In addition, the rotational stiffness and the moment resistance were measured under monotonic loading and compared with analytical predictions based on an analytical model (Stamatopoulos and Malo 2020a). The mock-up frame assembly was subjected to cyclic lateral loading and also tested with experimental modal analysis. The lateral stiffness, energy dissipation, fundamental eigenfrequencies and structural damping of the frame assembly were measured and quantified and compared with FE analyses. The following main conclusions are drawn:

  • The rotational stiffness of present connection measured from monotonic loading test was 4761 kNm/rad. The maximum moment resistance was 100.6 kNm limited by friction between the steel plates. No initial slip was recorded, and the connection demonstrated initial load take-up.

  • The rotational stiffness of connections for fully reversed loading and single-sided positive and negative moment tests were 4175, 4480 and 4900 kNm/rad, respectively. The coefficient of variation of stiffness values is from 4.0 to 8.6%, and hence, only minor differences are exhibited by this type of connection.

  • The mean rotational stiffness of connection from laboratory tests and from tests with mock-up frame assembly was on average 4175 kNm/rad and 4367 kNm/rad, respectively. Therefore, the measured values for connection properties obtained in the laboratory are also achievable in full-scale assemblies for this type of connection.

  • The rotational stiffness of connections obtained from fully reversed cyclic loading tests are very suitable for FE simulations. The FE model shows good agreement between the experimental results and the analytical model.

  • For unsymmetrical loading conditions when torsional rotation of the frame assembly is introduced, the transversal rotational stiffness of the connections should be considered. The transversal rotational stiffness value of these moment-resisting connections can be assumed to be around 0.20–0.30 \({K}_{\uptheta ,y}\).

  • The moment-resisting connections and mock-up frame assembly tests under cyclic loading showed high-energy dissipation; 4.2% and 3.5% in terms of equivalent viscous damping, respectively.

  • Cyclic energy dissipation from quasi-static component tests can be used to predict the structural damping in moment-resisting frames. Provided that the amount of material damping is known in the components, the damping caused by the connections can be modelled by equivalent viscous damping in FE models, and consequently the structural damping in framed structures can be predicted.