This technical note presents the experimental validation of a hybrid ﬁre testing coor- dination algorithm recently developed by some of the authors. For the ﬁrst time, the algorithm is applied to solve the static response of a multiple-degrees-of-freedom hybrid model. Keywords: Hybrid ﬁre testing, dynamic relaxation, partitioned time integration. 1. Introduction 1 Hybrid ﬁre testing (HFT) replaces a prototype structure under test with a hybrid model 2 that combines physical and numerical substructures (PS and NS, respectively). The 3 PS is tested in the laboratory using servo-controlled actuators and furnaces. The NS is 4 instantiatedinastructuralanalysissoftware. Acoordinationalgorithmensureskinematic 5 compatibility and force balance between PS and NS. There is a general consensus among 6 researchers that coordination algorithms should rely on a constant PS tangent stiﬀness 7 matrix estimate, possibly equal to its initial value, and facilitate the reuse of existing 8 structural analysis software for simulating the NS. Consistent with this philosophy, a 9 HFT coordination algorithm was recently proposed by Abbiati et al.[1], which relies on 10 dynamic relaxation and partitioned time integration. The experimental validation of the 11 algorithm on a single-degree-of-freedom (single-DoF) hybrid model was recently carried 12 out at the Danish Institute of Fire Safety and Security Technology by Sauca et al.[2]. 13 This technical note reports the results of the experimental validation of the same 14 HFT algorithm on a multiple-DoFs hybrid model. 15 2. Coordination algorithm 16 The HFT algorithm proposed in [1] combines the dynamic relaxation method and par- 17 titioned time integration to solve the static coupled equation of motion of the hybrid 18 model. The basic idea behind dynamic relaxation is to obtain the displacement solution 19 Corresponding author e-mail: abbiati@cae.au.dk Draft submitted to engrXiv DRAFT of a static structural problem by computing the transient response of an equivalent dy- 20 namic system. For a hybrid model comprising a single PS and a single NS, the system 21 of coupled equations of motion reads, 22 8 >>>>>>< >>>>>>:( MN uN+CN_ uN+rN uN =fN+LNTN LN_ uN+LN_ ug=0( MP uP+CP_ uP+rP uP =fP+LPTP LP_ uP+LP_ ug=0 LNTN+LPTP=0(1) where rare internal restoring force vectors while f(t)represent time-varying external loading; MandCare ﬁctitious mass and damping diagonal matrices, which are deﬁned as, Mii=(1:1t)2 4X jjKijj,Cii= 2! 0Mii (2) whereKijisagenericelementoftheinitialtangentstiﬀnessmatrix K=@r=@u;tisthe 23 timestepsizeoftheequivalenttransientanalysis; !0isthelowereigenvalueofthecoupled 24 equationsofequilibrium, asdeﬁnedin(1), whichisestimatedwithalinearizedmonolithic 25 version of the hybrid model. In order to cast compatibility and equilibrium equations 26 among substructures, LandL, which are Boolean signed matrices, localize interface 27 DoFs on every single substructure and vector ug, respectively. The latter gathers all 28 DoFs that are shared between at least two substructures, namely the interface DoFs. 29 3. Experimental validation 30 The experimental validation of the coordination algorithm relies on the 3-DoFs prototype 31 structure reported in Figure 1, which consists of a simply supported beam with elastic 32 restraints. Speciﬁcally, u1,u2andu3indicate the axial and the two rotational DoFs 33 of the hybrid model while r1,r2andr3, the corresponding restoring force and bending 34 moments. The PS coincides with the beam element and consists of an aluminium plate of 35 bh= 2002mm rectangular cross-section and L= 460mm length. The main material 36 parameters of aluminum are Young modulus E= 69:50 GPa, density = 2700kg/m3, 37 thermal expansion = 23106C-1, thermal conductivity = 201W/mC, speciﬁc heat 38 cp= 910J/kgC. The NS coincides with the elastic restraints, that is, the lateral spring of 39 stiﬀnessK1= 2500:00 kN/mm and both rotational spring of stiﬀness K2= 876:25 and 40 K3= 1052:50 kNmm/rad. Thermal loading is imposed to the PS by means of a 2:4kW 41 infrared lamp module. In detail, the temperature measured at the center of the plate is 42 linearly ramped from the room temperature 020C tomax= 135C in about 1400s. 43 The following hold phase lasts for 600s. Noteworthy, the stiﬀness of the axial spring K144 is about twenty times larger than the axial stiﬀness of the plate. As a result, the latter 45 tends to buckle if subjected to thermal expansion. 46 Figure 2 provides an overview of the experimental setup, for which a comprehensive 47 description is reported in [3]. 48 2 DRAFT u2u1 K3 θ(t)E,I,A,L,ρ,α,λ,cpK2 K1 u3 Figure 1: Validation case study: a) prototype structure; b) temperature history. Figure 2: Experimental setup: a) 3-DoFs test rig; b) plate specimen; c) infrared lamp. 4. Results and discussion 49 The eﬀectiveness of the algorithm is demonstrated by examining response quantities 50 related to a single HFT. In this regard, Figure 3 compares PS and NS displacement 51 response quantities while Figure 4 compares PS and NS force response quantities. For 52 the sake of comparison, in the latter case, PS response quantities are multiplied by 1 53 so that matching with NS response quantities can be graphically veriﬁed. As one can 54 note, some discontinuity characterizes both u3andr3, which are associated with the 55 right rotational DoF of the hybrid model. The reason is that the bearings of the right 56 clevis of the 3-DoFs test rig are not perfectly aligned and the joint suﬀers from stick and 57 slip behavior as reported in [3]. However, as can be observed, both compatibility and 58 balance between PS and NS are attained with a high degree of accuracy. Finally, for the 59 sake of comparison, residual forces, which are computed as the sum of ﬁctitious inertia 60 and damping forces of both PS and NS, are reported in Figure 5. As can be observed, 61 residual forces are small compared to restoring forces of Figure 4. 62 (a) (b) (c) Figure 3: Displacement response a) displacement DoF u1; b) rotation DoF u2; c) rotation DoF u3. 3 DRAFT (a) (b) (c) Figure 4: Restoring forces: a) displacement DoF u1; b) rotation DoF u2; c) rotation DoF u3. (a) (b) (c) Figure 5: Residual forces: a) displacement DoF u1; b) rotation DoF u2; c) rotation DoF u3. 5. Conclusions and outlooks 63 Experimental results demonstrate that, with the proposed coordination algorithm, com- 64 patibility and balance between physical and numerical substructures are attained with a 65 high level of accuracy. Also, residual forces are small compared to restoring forces. Hy- 66 brid ﬁre testing with nonlinear numerical substructures will be the target of our future 67 work. 68 Acknowledgments 69 The ﬁrst and third authors were supported by the Marie Skłodowska-Curie grants 764547 70 and 813424, respectively. The 3-DoFs test rig was funded by the Swiss Space Oﬃce, 71 THERMICS MdP2016 Project, grant REF-1131-61001. 72