Experimental evaluation of the lateral load distribution in the elastic-plastic phase of timber-steel hybrid structures with a novel light timber-steel diaphragm

Abstract

A cyclic loading experiment involving a timber-steel hybrid structure consisting of a steel frame and a novel light timber-steel diaphragm is presented to quantify the flexibility of the diaphragm and its ability to distribute lateral loads in the elastic-plastic phase of the structure. A lateral load-distribution factor was proposed, and its relationship to the ratio of the stiffness of the diaphragm to that of the lateral load-resisting elements was investigated. The diaphragm was classified based on these variables. The results indicated that the failure modes of the structure were associated with the forms of damage experienced by the lateral load-resisting elements, whereas little damage was observed for the diaphragm. The diaphragm exhibited the ability to continuously adjust the distribution of lateral loads to each lateral load-resisting element; accordingly, each lateral load-resisting element had approximately the same shear force, the same lateral stiffness, and the same lateral displacement during the loading process. As the lateral displacement increased, the stiffness ratio and load-distribution factor both gradually increased, and the diaphragm correspondingly changed from semi-rigid to rigid. At times, as the lateral displacement increased, the diaphragm rapidly became rigid, and it was unnecessarily rigid during the initial loading phase when the in-plane stiffness reached a certain threshold.

Keywords

in-plane stiffness lateral load distribution lateral load-resisting element light timber-steel diaphragm timber-steel hybrid structure

Introduction

A novel light timber-steel diaphragm is depicted in Figure 1. C-shaped steel members constitute the joists. Spruce-pine-fir (SPF) dimensional lumber is laid perpendicular to the joists and attached using screws to form a wooden deck. Subsequently, steel mesh is fixed to the wooden deck via U-staples. Finally, lightweight polyester mortar is poured as the top layer to strengthen the in-plane and flexural stiffness of the diaphragm. The dead weight of the diaphragm is approximately 800 kg/m³, which is approximately one-third the dead weight of concrete diaphragms. Timber-steel hybrid structures composed of these diaphragms with steel frames boast many advantages, including a notable energy conservation potential, a high degree of prefabrication, a light weight, and a good seismic performance.

Figure 1.

Schematic diagram of the light timber-steel diaphragm.

Diaphragms distribute lateral loads to lateral load-resisting elements (LLREs) and enable them to cooperate with each other to resist earthquakes. The in-plane stiffness of a diaphragm influences its capacity to distribute lateral loads, thereby affecting the seismic performance of the structure. If the diaphragms are rigid, then loads are distributed to LLREs according to their lateral stiffness values; in contrast, if the diaphragms are flexible, then they are distributed to LLREs based on their respective tributary area or influence area. A concrete diaphragm, which exhibits remarkable in-plane stiffness, can effectively transfer loads and improve the seismic performance of structures. Conversely, Chinese traditional wood diaphragms are flexible; therefore, loads cannot be transferred, and thus, the seismic performance of structures containing these diaphragms is poor.

The screws, U-staples, and polyester mortar used to construct the diaphragm presented in this article strongly affect its flexibility and its ability to transfer loads. The classification of the diaphragm flexibility depends on the ratio of the in-plane stiffness of the diaphragm to the lateral stiffness of the associated LLREs. American Society of Civil Engineers (ASCE) 7-10 (2010) states that diaphragms are considered flexible when the maximum in-plane deflection is more than twice the average inter-story drift; otherwise, they are considered rigid. However, no such definition is given for semi-rigid diaphragms. ASCE 41-06 (2006) presents the same definition for flexible diaphragms as does ASCE 7-10, but ASCE 41-06 classifies diaphragms as rigid if the ratio of the maximum in-plane deformation to the average inter-story drift is below 0.5; in addition, diaphragms are regarded as semi-rigid if the ratio is between 0.5 and 2.0. Most other design standards do not provide guidance with regard to classifying such diaphragms. Pecce et al. (2017, 2018) evaluated the effects of some parameters (e.g. the typologies of LLREs and building geometry) on the in-plane deformability of reinforced concrete (RC) floors. The floor deformability was examined by means of a stiffness index R_i or R_i,wall and by measuring the error with respect to the rigid floor assumption by two displacement ratio parameters: δ₁/δ₂ and δ_rig/δ_flex. Finally, whether a rigid assumption of RC floors should be adopted in various cases was discussed.

Few studies have proposed methods for calculating the load distributions of semi-rigid diaphragms; instead, they presented only relative approximations. As shown in Chinese Standard (GB50011, 2010), if a diaphragm is semi-rigid, then the distributed load is calculated as the average of the assumed rigid and flexible diaphragm flexibilities. However, the application of this method is inaccurate and sometimes dangerous. Similarly, for semi-rigid diaphragms, design methods that adopt the maximum force of the two diaphragm flexibility assumptions (Association of Professional Engineers and Geoscientists of British Columbia (APEGBC), 2015) may be unsafe, as noted by Chen et al. (2014).

Recently, many studies have focused on the in-plane stiffness of single straight-sheathed wood diaphragms, to which the diaphragm considered in this article is similar (Baldessari, 2010; Branco et al., 2015; Giongo et al., 2015; Li et al., 2016; Newcombe et al., 2010; Valluzzi et al., 2016; Wilson et al., 2014). However, few publications are available regarding the influence of the diaphragm flexibility on the load distribution for structures in the elastic-plastic phase. A multiple spring model was proposed by Chen et al. (2014) to investigate the load distribution from the diaphragms to the LLREs in timber structures with various stiffness ratios. Li et al. (2010) performed an experiment on a hybrid structure composed of a concrete frame and timber diaphragms under horizontal forces; the results demonstrated that the diaphragm was close to being rigid in the elastic-plastic phase of the structure. Nevertheless, the investigation neither considered a classification of the diaphragm flexibility nor determined the loads transferred.

This article presents a destructive experiment involving a timber-steel hybrid structure with a novel light timber-steel diaphragm to evaluate the flexibility of the diaphragm with a new method and to quantify the effect of the diaphragm on the load distribution in the overall elastic-plastic phase of the structure.

Experimental methods

Test specimen

As depicted in Figure 2, the specimen was a single-story by two-span timber-steel hybrid structure composed of a steel frame and a novel light timber-steel diaphragm in addition to light wood–framed shear walls. The structural length, width, and height were 6.0, 3.0, and 2.8 m, respectively. The flanges and webs of the steel main beams were welded and bolted to steel columns, respectively. The steel link beams were bolted to steel plates welded onto the column webs. Hot-rolled H-section members made of mild carbon steel Q235B (GB50017, 2003), H-150 × 100 × 6 × 9 mm, were used for the main beams and link beams, and H-150 × 150 × 7 × 10 mm steel was used for the columns. The diaphragm was bolted to the main beams to form a lateral load-distribution system (LLDS), whereas the lateral load-resisting system was composed of three identically constructed LLREs separately named RF-1, RF-2, and RF-3. Each RF was composed of a light wood–framed shear wall in a single steel frame and was secured by bolts. The mechanical properties of the materials are listed in Table 1.

Figure 2.

Test specimen.

Table 1.

Mechanical properties of the test materials (units: MPa).

Member and material			Yield strength	Maximum strength	Elastic modulus
Steel frame	Main (link) beam		260	400	1.9 × 10⁵
	Column		270	430	1.9 × 10⁵
Light timber-steel diaphragm	Steel joist		290	410	2.0 × 10⁵
	Wood deck	Parallel to grain	/	39	12,500
		Perpendicular to grain	/	4	310
	Wood screw		330	440	1.9 × 10⁵
	Bar-mat reinforcement		235	370	2.1 × 10⁵
	U-staple		210	340	2.0 × 10⁵
	Topping		/	21	2.0 × 10⁴
Light wood–framed shear wall	Wood stud	Parallel to grain	/	45	15,300
		Perpendicular to grain	/	6	390
	OSB panel	Parallel to shavings	/	23	5530
		Perpendicular to shavings	/	9	1640
	Nail	Stud-to-OSB (stud)	851 (790)	/	/
Fasterner between the wood walls and steel frame			640	800	2.0 × 10⁵

OSB: oriented strand board.

LLDS

The layout of the LLDS and its cross-section are depicted in Figure 3. The length, width, and height of the diaphragm were 6.0, 3.0, and 0.126 m, respectively. Figure 4 shows the field construction. Two 2895-mm long C-section steel members, C-160 × 50 × 20 × 2.5 mm, were placed back to back and arranged perpendicular to the loading direction to form the diaphragm joists. The spacing between the side joists and the link beams was 230 mm and that between the side and the intermediate joists was 370 mm, while the spacing between the intermediate joists was 450 mm (Figures 3(a)). SPF dimensional lumber with a 38 × 184 mm cross-section and a length of 2900 mm was perpendicularly connected to the joists by using $ϕ$ 4.0 mm × 35 mm threaded screws to form the deck. The joists were predrilled with ϕ 5.2-mm holes to facilitate the installation of the screws; the intermediate and side spacing values of the screws on each piece of SPF lumber along the longitudinal direction of the joists were 124 and 30 mm, respectively (Figure 4(a)). Next, a ϕ 4.0 mm × 100 mm steel mesh with ribbed wires was fixed onto the deck using staggered arranged ϕ 3.0 mm × 32 mm U-staples at the intersections of the steel mesh and then spot-welded to the steel beams (Figures 4(b)). Subsequently, a 30-mm thick layer of lightweight polyester mortar was paved as the top layer, thereby completing the construction of the diaphragm (Figure 3(b)). Finally, the diaphragm was bolted to the main beams to form the LLDS by a total of 28 T-shaped connectors, which were composed of a plate steel perpendicularly welded to angle steel. The plate steel of the T-shaped connectors was inserted into the gap between the two C-shaped steel members and then bolted together by two M14 bolts, whereas the angle steel of the T-shaped connectors was fixed onto the bottom flange of the main beams by two M14 bolts on the flange and two M14 bolts on the web of main beams (Figures 3(b) and 4(a)).

Figure 3.

LLDS (mm): (a) layout and (b) 1:1 section.

Figure 4.

Field test construction of the LLDS: (a) steel joists and SPF lumber deck (mm) and (b) steel mesh with ribbed wires and U-staples.

Note that no waterproof seals (Figure 3(b)) were used as gap covers for the bottom steel reinforcement shown in Figure 4 because they were soft and had little influence on the mechanical properties of the diaphragm.

LLRE

The details regarding RF-1 are depicted in Figure 5(a). The height and width of the single steel frame were 2.8 and 3.0 m, respectively. An in-filled wood-framed shear wall with a height of 2.526 m and a width of 2.850 m overlaid by 14.68-mm thick oriented strand board (OSB) on two sides of the wood frame was connected to the steel frame by bolts to form RF-1. The wood frame was composed of wood studs and double blockings in addition to a double top plate and a single bottom plate, all of which were composed of SPF lumber with a 38 × 140 mm cross-section. The studs spaced 406-mm apart were connected to the blockings and to the top and bottom plates by ϕ 3.3 × 64 mm spiral nails to form the wood frame. Then, two types of OSB sheathings with dimensions of 1625 × 842 and 1225 × 842 mm (length × width) were connected to the wood frame by ϕ 3.3 × 82 mm spiral nails to form the shear wall; the intermediate nail spacing along the longitudinal direction of the interior studs was 300 mm, and the side nail spacing on the top and bottom plates, blockings, and double-end studs was 150 mm. The top plate was connected to the bottom flange of the main beam by a total of 14 M14 bolts arranged in two bolts per row with a spacing of 360 mm. The end studs were connected to the flanges of the columns by a total of 12 M14 bolts arranged in two bolts per row with a spacing of 406 mm. More details regarding the LLRE can be found in studies of He et al. (2014) and Li et al. (2014).

Figure 5.

LLRE (RF-1): (a) layout of the strain gauges and shear force shared between the steel frame and the in-filled wood-framed shear wall (mm) and (b) calculation of the shear force of the left steel column.

Test setup and loading scheme

The test setup is shown in Figure 6. On one side, two separate distributive beams, namely, DB-1 and DB-2, were both hinged with a jack at their mid-span; on the other side, the ends of the two beams were separately hinged to the tops of the side and middle columns. Specifically, the two ends of DB-1 were hinged to points 1 and 3, and the two ends of DB-2 were hinged to points 3 and 5. Therefore, the two beams can rotate freely following structural deformation. In addition, the ratio of the concentrated lateral load of the middle LLRE to that of the side LLRE was 2:1 (Figure 2). As depicted in Figure 7, a cyclic loading protocol controlled by the total force of the specimen and the lateral displacement of RF-2 before and after the elastic structural bearing capacity predicted by numerical methods, was adopted with one loop for each load step and three loops of uniform amplitude for each displacement level.

Figure 6.

Test setup.

Figure 7.

Loading scheme.

Data acquisition

The lateral displacements of the LLREs were measured by LVDTs (Figure 2). The shear forces of the LLREs were derived from the measurements acquired by strain gauges arranged along the flanges of the steel frame (Figure 5(a)).

Taking RF-1 as an example, the shear force on each LLRE (Q_RF) was shared by the in-filled wood wall (Q₃) and the steel frame (Q_SF), where Q_SF is the sum of the shear forces in the left column (Q₁) and right column (Q₂).

Figure 5(b) illustrates the method used to calculate Q₁ (or Q₂). Strain gauges S3–S6 were located on the 1-m long segment of the left column. This segment was close to the inflection point of the steel frame and was thus kept elastic during the loading process according to numerical analysis. He et al. (2014) provided a formula for calculating Q₁ (or Q₂), as described by equation (1)

Q_{1 or 2} = \frac{M_{t} + M_{b}}{L} = \frac{(ε_{t max} + ε_{t min} + ε_{b max} + ε_{b min}) EW}{2 L}

(1)

where M_t (M_b) is the bending moment on the top (bottom) section of the segment; ε_tmax (ε_tmin) and ε_bmax (ε_bmin) are the maximum (minimum) strain of the top and bottom section, respectively, of the segment; E is the elasticity modulus of steel; W is the section modulus of the column; and L is the length of the elastic segment.

The value of Q_SF, namely, Q₁ + Q₂, was calculated by equation (1), and the proportion of Q_SF distributed to each single steel frame was the same as that for Q_RF distributed to each LLRE. Therefore, it is easy to obtain the proportion of Q_RF distributed to each LLRE. In addition, the total load applied to the specimen, namely, 4P, was recorded by the jacks. Consequently, the Q_RF of each LLRE was obtained based on the proportion of Q_RF distributed to each LLRE and the recorded total load.

Results and discussion

The hysteretic loop and envelope curves of each LLRE are shown in Figure 8, in which the curves are given in terms of the lateral displacement and the shear force of each LLRE.

Figure 8.

Hysteretic curves of (a) RF-1, (b) RF-2, and (c) RF-3.

Failure modes

The lateral displacements of the LLREs were almost equal when the displacement of RF-2 was less than approximately 100 mm. However, when the displacement increased above it, the structure experienced torsion and dissymmetry due to the occurrence of a weld fracture between the main beam and the column in RF-3. At that moment, the inter-story drift was 1/29, that is, far larger than 1/50, which is the failure limit for steel structures defined in Chinese Standard GB50011 (2010). Therefore, the characteristics of the specimen after damage will not be considered in the following discussion. The main failure modes of the structure encompassed the forms of damage experienced by the LLREs, including the fatigue fracturing of the nails in the wood-framed walls and the weld fractures in the steel frame (Figure 9(a)). However, few serious instances of damage were observed in the LLDS. The polyester mortar was only locally crushed at the loading point of the diaphragm, and in other regions, only some microcracks with a width of about 1-mm wide were found (Figure 9(b)).

Figure 9.

Failure modes: (a) fatigue fracturing of nails and weld fractures and (b) microcracks and locally crushed polyester mortar.

Theory regarding the effect of the diaphragm flexibility on the lateral load distribution

The in-plane stiffness of the LLDS includes the in-plane stiffness of the diaphragm and the shear stiffness of the connections between the diaphragm and the beams. If the connections are rigid or if their stiffness values are sufficiently large, the deformation of the LLDS is related only to the diaphragm flexibility and the in-plane stiffness of the LLDS is equivalent to that of the diaphragm alone (Brignola et al., 2009). As mentioned above, the experiment included a total of 28 T-shaped connectors for the diaphragm, and each connector was bolted to the beams by four M14 bolts; thus, such a large number of bolts provided a sufficiently large shear stiffness compared with the in-plane stiffness of the diaphragm. Furthermore, little slippage was observed at the connections during the test; consequently, the connections are considered rigid and the load distribution is influenced only by the diaphragm flexibility in this article. To quantify the ability of the diaphragm to distribute loads, the lateral load-distribution factor, β, is proposed, as described by equation (2)

β = \frac{P_{tra}}{P_{mid} - P_{tot} / 3} = \frac{P_{tra}}{2 P - (P + 2 P + P) / 3}

(2)

where P_mid is the load on RF-2, and its value is 2P; P_tot is the total load on the specimen, and its value is P + 2P + P; and the numerator P_tra is the total load transferred from RF-2 to RF-1 and RF-3 at any moment within the test structure by the presented hybrid diaphragm during the loading process, and it can be obtained by subtracting the measured shear force of RF-2 from P_mid, namely, 2P–Q_RF-2.

If the diaphragm is rigid, the value of P_tra is 2P/3. This is because each LLRE was identically constructed with the same lateral stiffness values; in addition, the shear force of each LLRE is distributed based on its lateral stiffness in a structure with a rigid diaphragm. Therefore, the shear force of each LLRE is P_tot/3, namely, 4P/3, and P_tra is equal to P_mid–(P_tot)/3, namely, 2P/3. Thus, the denominator in equation (2) denotes the load transferred from RF-2 to RF-1 and RF-3 when the diaphragm is rigid. Based on the above discussion, $β$ represents the ratio of the load transferred by a diaphragm with an actual in-plane stiffness at any moment to that transferred by a rigid diaphragm.

The value of β is influenced by the ratio of the in-plane stiffness of the diaphragm, k_d, to the lateral stiffness of the LLRE, k_RF, denoted as α, namely, k_d/k_RF. Equation (2) shows that if the diaphragm is flexible, then α is 0 and the load on RF-2 cannot be transferred to RF-1 and RF-3; therefore, $P_{tra}$ and β are both equal to 0. If the diaphragm is rigid, then α mathematically tends to infinity; thus, $P_{tra}$ is $2 P / 3$ , and β is 1. If the diaphragm is neither rigid nor flexible, then β is between 0 and 1. As β increases, the load is increasingly distributed from RF-2 to RF-1 and RF-3. Hence, the value of $β$ reflects the extent of the distribution of a lateral load by the diaphragm and is strongly related to α.

It is difficult to directly obtain the value of k_d and α experimentally. However, k_d can be calculated by equation k_d =αk_RF, where k_RF can be easily obtained by dividing the measured Q_RF by the measured lateral displacement of each LLRE, δ_RF, namely, k_RF = Q_RF/δ_RF. β can also be easily calculated by equation (2), and it has a one-to-one correspondence with α. Therefore, understanding the α–β relationship is crucial for obtaining α and k_d.

To investigate the α–β relationship of the symmetrical test structure, a finite-element analysis was performed using ABAQUS 6.9 software, as depicted in Figure 10. The joints connecting the main beams to the columns were idealized as rigid, and the joints connecting the link beams to the columns were assumed to be flexible. All beams and columns were modeled by S4R elements (shell elements). The mechanical properties of the steel are listed in Table 1. The diaphragm was modeled by a pair of diagonal springs (He et al., 2011; Li et al., 2010; Xu and Dolan, 2009) with SPRING2 (linear spring) elements hinged with steel frames to simulate the in-plane stiffness, k_d (i.e. the in-plane stiffness of the diaphragm with a concentrated load acting on its end). The stiffness of the SPRING2 elements, k_s, can be inferred by k_d (He et al., 2011), as given by equation (3)

k_{s} = \frac{k_{d}}{2} \times \frac{L^{2} + H^{2}}{H^{2}}

(3)

where L and H are the span and width of the diaphragm, respectively.

Figure 10.

Numerical model of the test structure.

The wood-framed shear walls and the bolt connections between them and the steel frame were also modeled by SPRING2 elements, but the stiffness values of the SPRING2 elements were all assumed to be 0 to reduce the computation time. This is because the ratio, k_d/k_RF, constitutes a key factor when obtaining the α–β relationship rather than the actual stiffness value of each LLRE, k_RF. Therefore, k_RF was actually the lateral stiffness of the single steel frame in this case. By keeping k_RF unchanged and changing the value of k_s, α varied accordingly, and $β$ was obtained via equation (2).

Table 2 describes the α–β relationship obtained via the numerical simulation in addition to the values of Q_RF-1/Q_RF-2 (or Q_RF-3/Q_RF-2), which denotes the corresponding ratio of the shear force on the side LLRE to that on the middle LLRE.

Table 2.

The $α - β$ relationship.

$α$	1:10	1:5	1:4	1:3	1:2	1:1	2:1	3:1	4:1	5:1	10:1
Original $β$ (%)	22.9	36.9	42.1	49.0	58.6	72.8	82.8	86.8	88.9	90.3	93.1
Fitting $β$ (%)	24.8	38.2	42.5	48.0	55.9	69.3	82.7	90.5	90.9	91.3	93.1
$\frac{Q_{RF - 1}}{Q_{RF - 2}}$ $(\frac{Q_{RF - 3}}{Q_{RF - 2}})$	0.583	0.640	0.663	0.695	0.743	0.820	0.881	0.907	0.921	0.930	0.950

The results of the α–β relationship based on Table 2 and its fitting curve are shown in Figure 11. The fitting curve was divided into two parts: a logarithmic segment for $0 < α \leq 3$ given by $β$ = 0.193ln( $α$ ) + 0.693 and a linear segment for $α > 3$ given by $β$ = 0.894 + 0.0037 $α$ . The regression coefficient R² for the logarithmic segment and for the linear segment is 0.994 and 0.999, respectively, so the fitting curve exhibits good agreement with the original data point. The fitting values of β are listed in Table 2. It is worth noting that the trend of the curve is similar to those of the curves relating the stiffness index R_i (R_i,wall) to the displacement ratio δ₁/δ₂ (δ_rig/δ_flex) presented by Pecce et al. (2017, 2018); this is because, the shear force and the lateral displacements of the LLREs are strictly related.

Figure 11.

The $α - β$ curve.

As shown in Figure 11, $β$ increasingly tended to 100% as α increased. When $α$ ranged from 0 to 3, $β$ rapidly increased from 0% to 86.8%; additionally, when $α$ ranged from 3 to 10, $β$ increased by only 6.3%. Accordingly, the diaphragm can be considered rigid when $α > 3$ because its capacity to distribute the load was sufficiently large; moreover, improving the in-plane stiffness of the diaphragm had no obvious influence on the load distribution in this case. When $α$ was less than 0.2, $β$ was less than 36.9%. In this case, the diaphragm can be regarded as flexible because the capacity of the diaphragm to distribute the load was very low. When $0.2 \leq α \leq 3$ , $β$ varied between 36.9% and 86.8%. In this case, the diaphragm can be classified as semi-rigid, emphasizing that the in-plane stiffness values of the diaphragm should be considered during structural analysis.

Nonlinearity of the lateral load distribution

To examine the nonlinearity associated with the load distribution during the overall elastic-plastic phase of the structure, the lateral displacements and shear force ratios among the LLREs were considered during forward loading, as shown in Figure 12.

Figure 12.

Lateral displacement and shear force ratios.

Generally, the shear force ratios of Q_RF-1/Q_RF-2 and Q_RF-3/Q_RF-2 both increased from 0.85 to approximately 1.0, suggesting that increasing loads were transferred from RF-2 to RF-1 and RF-3 and that the diaphragm became increasingly rigid. The reason for this behavior was that α continuously increased as the displacement increased because the stiffness degradation rate of the LLREs was far higher than that of the diaphragm (Figures 13 and 16). Figure 12 also shows that when the displacement of RF-2, δ_RF-2, was below approximately 15 mm, Q_RF-1/Q_RF-2 and Q_RF-3/Q_RF-2 changed rapidly; however, when δ_RF-2 was greater than it, these ratios slowly varied. This finding is mainly attributed to the fact that $β$ initially increased sharply and then gradually increased toward the end of the test as $α$ increased (Figure 11).

Figure 13.

The secant stiffness degradation curves of (a) the first envelope curve of each LLRE and (b) each envelope curve of RF-2.

As shown in Figure 12, the lateral displacement ratios of δ_RF-1/δ_RF-2 and δ_RF-3/δ_RF-2 were approximately 0.85 after the initial loading stage, and they subsequently increased to approximately 1.0 with fluctuations within a relatively small range. This phenomenon can be explained as follows. First, although the lateral stiffness of each LLRE was the same before the initial loading, the displacement of RF-1 (RF-3) was only approximately 85% of that of RF-2 after loading because, at that moment, the diaphragm was not rigid and was consequently unable to transfer all of the redundant force on RF-2, namely, $2 P / 3$ , and was unable to ensure that the displacement was equal for all LLREs. Given that RF-1 initially exhibited a larger shear force and a larger displacement than RF-3 because of the loading process or other reasons, the stiffness of RF-1 gradually decreased to a value less than that of RF-3 because the stiffness of each LLRE was nonlinear and negatively related to the load. When the structure was loaded and the degree of displacement increased, the diaphragm eventually distributed smaller loads to RF-1 and larger loads to RF-3. Therefore, the differences in the shear force, stiffness, and displacement between RF-1 and RF-3 gradually decreased. When the stiffness of RF-1 was gradually adjusted to a value greater than that of RF-3, the opposite situation occurred. In conclusion, the diaphragm constantly influenced the shear force, lateral stiffness, and lateral displacement of the LLREs and allowed each LLRE to have nearly the same shear force, the same lateral stiffness and the same lateral displacement in the elastic-plastic phase of the structure, thereby improving the cooperative effects among the LLREs attributable to the increasingly distributed shear force.

Figure 12 also shows that δ_RF-1/δ_RF-2 (δ_RF-3/δ_RF-2) increased to approximately 1.0 more rapidly than did Q_RF-1/Q_RF-2 (Q_RF-3/Q_RF-2), and their curves do not completely coincide. This is because the displacement is the ratio of the distributed shear force to the stiffness, and there was a positive correlation between the force and stiffness. Specifically, the quantity of the shear force distributed to each LLRE depends on the lateral stiffness in the case of the diaphragm with an actual stiffness. An LLRE with a larger stiffness would also bear a greater shear force; thus, the displacements of the LLREs equalize more rapidly than do the shear forces. Therefore, evaluating the diaphragm flexibility depends on the magnitude of the transferred shear force rather than the displacement ratio among the LLREs because, similar to a rigid diaphragm, a semi-rigid diaphragm might quickly equalize the displacements among all of the LLREs.

Quantifying the capacity of the diaphragm to distribute lateral loads

Based on Figure 8, the secant stiffness degradation curves of the first envelope curve of each LLRE and of each envelope curve of RF-2 are shown in Figure 13. The lateral stiffness values of the LLREs were slightly different at any given moment (Figure 13(a)), which confirms the assumption that they had the same stiffness, as discussed above.

According to Figure 8, the $β$ curves computed by equation (2) for each envelope curve are shown in Figure 14. The $β$ curve for the first envelope curve was fitted very well by two segments (Figure 14). The first segment is logarithmic at δ_RF-2 ≤ 15 mm given by $β$ = 0.066ln(δ_RF-2) + 0.721. The other segment is linear at δ_RF-2 > 15 mm given by $β$ = 0.890 + 0.00033δ_RF-2. As δ_RF-2 increased, the $β$ curves steadily increased from 0.75 to approximately 1.0. This finding indicates that $α$ continuously increased and that the diaphragm changed from semi-rigid to rigid. The boundary between these states was reached when δ_RF-2 was approximately 15 mm because, at that moment, $β$ was approximately 86.8%, which represents the limit for classifying the two diaphragm assumptions, as previously discussed.

Figure 14.

The $β$ curves.

Figure 14 also shows that $β$ rapidly increased at δ_RF-2 values below approximately 15 mm and slowly increased above it. This observation can be explained as follows. First, the in-plane stiffness of the diaphragm only slightly declined (Figure 16). Second, the stiffness values of the LLREs dramatically decreased for displacements below approximately 15 mm and only gradually changed for displacements exceeding it (Figure 13(a)). Therefore, $α$ behaved in a manner similar to that of the stiffness values of the LLREs. Finally, $β$ was sensitive to $α$ in the initial phase but insensitive in the later stage (Figure 11).

Note that the $β$ curve for the first envelope curve was slightly smaller than those for the second and third curves (Figure 14). This difference suggests that the stiffness values of the LLREs decreased in the second and third loading cycles compared with those in the first cycle (Figure 13(b)).

Diaphragm flexibility

Based on the α–β relationship (Figure 11), the $α$ curve derived from the $β$ fitting curve (Figure 14) is shown in Figure 15. Furthermore, the in-plane stiffness (k_d) curve of the diaphragm shown in Figure 16 is derived from the $α$ curve (Figure 15) and the average of the stiffness degradation curves (Figure 13(a)).

Figure 15.

The $α$ curve.

Figure 16.

The in-plane stiffness curve of the diaphragm.

As seen in Figure 15, similar to those of the $β$ curves, the trend of the $α$ curve also increased continuously as the displacement increased, with an initial period of rapid change. The value of $α$ was approximately 1.3 at the initial time of loading, which indicates that the diaphragm was semi-rigid at that time. However, as δ_RF-2 increased to approximately 15 mm, the value of $α$ increased to approximately 3.0. At that moment, the inter-story drift of the structure was approximately 1/187, that is, slightly larger than 1/250, which is the elastic limit for steel structures specified in Chinese Standard GB50011 (2010). Thus, the structure reached the early plastic state. Subsequently, the diaphragm was rigid. Note that the diaphragm did not always have to transfer 100% of the lateral load in the initial loading period to create an equal shear force for each LLRE. The value of α increased beyond 3.0 only when the loads transferred by the diaphragm reached a certain quantity, as will be discussed; at this moment, the diaphragm rapidly transformed from semi-rigid to rigid. Figure 16 shows that k_d slightly declined in the elastic-plastic phase of the structure. Note that neither $α$ nor k_d was accurately estimated as the displacement increased. For example, k_d was incorrectly predicted to increase when δ_RF-2 was greater than 60 mm because $β$ was sufficiently large at that moment, that is, β is not sensitive to $α$ (Figure 11). As a result, the measurement error increased the inaccuracy of both the estimate of $α$ and k_d. Fortunately, in this case, the inaccuracies of these variables mattered little during the structural design because the diaphragm was already rigid.

Conclusion

The flexibility of a novel light timber-steel diaphragm and its capacity to distribute lateral loads in the overall elastic-plastic phase of a timber-steel hybrid structure were investigated via a test. The main findings are as follows:

To evaluate the extent of loads distributed by the diaphragm, the lateral load-distribution factor $β$ for the test structure was proposed. This factor is strongly related to $α$ , which is the ratio of the in-plane stiffness of the diaphragm to the lateral stiffness of the LLRE. The diaphragm is considered rigid when $α$ > 3 or $β$ > 86.8%, flexible when $α$ < 0.2 or $β$ < 36.9%, and semi-rigid when $0.2 \leq α \leq 3$ or 36.9% $\leq β \leq$ 86.8%.

The ultimate failure modes of the structure were associated with the forms of damage experienced by the LLREs, whereas little damage was observed for the diaphragm during the entire elastic-plastic phase of the structure.

In the loading process, the diaphragm was able to continuously adjust the distribution of the lateral load to each LLRE, that is, so that each LLRE has approximately the same shear force, the same lateral stiffness and the same lateral displacement. This adjustment process was nonlinear and improved the cooperative effects among the LLREs.

As the lateral displacement of the structure increased, the value of $β$ and $α$ both continuously increased, and the diaphragm changed from semi-rigid to rigid. At times, the diaphragm rapidly becomes rigid only if the in-plane stiffness of the diaphragm is sufficiently large in the initial stage so that the transferred loads reach a certain threshold quantity; this phenomenon must be studied further.

For structures containing the presented diaphragm, redundant loads on the middle LLREs with large laterally loaded areas might not be completely transferred by the diaphragm. Therefore, the design strength of middle LLREs should be properly amplified to prevent them from being damaged before the diaphragms can become rigid.

Footnotes

Declaration of Conflicting Interests

The author(s) declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by the National Natural Science Foundation of China (Grant Nos.: 51608160 and 51378382), and the Industry–University–Research Funds for Hefei University of Technology (Grant No.: XC2016JZBZ09).

ORCID iD

Minjuan He

References

Association of Professional Engineers and Geoscientists of British Columbia (APEGBC) (2015) Structural, Fire Protection and Building Envelope Professional Engineering Services for 5 and 6 Storey Wood Frame Residential Building Projects (Mid-Rise Buildings), APEGBC Technical and Practice Bulletin, British Columbia, Canada: Association of Professional Engineers and Geoscientists of British Columbia.

ASCE 07-10 (2010) Minimum Design Loads for Buildings and Other Structures. Reston, VA: ASCE.

ASCE 41-06 (2006) Seismic Rehabilitation of Existing Buildings. Reston, VA: ASCE.

Baldessari

(2010) In-plane behaviour of differently refurbished timber floors. PhD Thesis, University of Trento, Trento.

Branco

Kekeliak

Lourenço

(2015) In-plane stiffness of traditional timber floors strengthened with CLT. European Journal of Wood and Wood Products 73(3): 313–323.

Brignola

Pampanin

Podestà

(2009) Evaluation and control of the in-plane stiffness of timber floors for the performance-based retrofit of URM buildings. Bulletin of the New Zealand Society for Earthquake Engineering 42(3): 204–221.

Chen

Chui

et al . (2014) Load distribution in timber structures consisting of multiple lateral load resisting elements with different stiffnesses. Journal of Performance of Constructed Facilities 28(6): A4014011.

GB50011 (2010) Code for Seismic Design of Buildings. Beijing, China: China Standard.

GB50017 (2003) Code for Design of Steel Structures. Beijing, China: China Standard.

10.

Giongo

Dizhur

Tomasi

et al . (2015) Field testing of flexible timber diaphragms in an existing vintage URM building. Journal of Structural Engineering 141(1): D4014009.

11.

Guo

et al . (2011) The seismic performance in diaphragm plane of multi-storey timber and concrete hybrid Structure. In: Proceedings of the 12th East Asia-Pacific conference on structural engineering and constructionHong Kong, 26–28 February.

12.

Lam

et al . (2014) Experimental investigation on lateral performance of timber-steel hybrid shear wall systems. Journal of Structural Engineering 140(6): 04014029.

13.

Guo

et al . (2010) Lateral load-bearing capacity of wood diaphragm in hybrid structure with concrete frame and timber floor. In: Proceedings of the 11th World conference on timber engineeringTrento, 20–24 June.

14.

Lam

et al . (2014) Finite element modeling and parametric analysis of timber-steel hybrid structures. The Structural Design of Tall and Special Buildings 23(14): 1045–1063.

15.

et al . (2016) In-plane behavior of timber-steel hybrid floor diaphragms: experimental testing and numerical simulation. Journal of Structural Engineering 142(12): 04016119.

16.

Newcombe

Van Beerschoten

Carradine

et al . (2010) In-plane experimental testing of timber-concrete composite floor diaphragms. Journal of Structural Engineering 136(11): 1461–1468.

17.

Pecce

Ceroni

Maddaloni

et al . (2017) Assessment of the in-plane deformability of RC floors with traditional and innovative lightening elements in RC framed and wall structures. Bulletin of Earthquake Engineering. 15: 3125–3149.

18.

Pecce

Ceroni

Maddaloni

(2018) In-plane deformability of RC floors: assessment of the main parameters and influence on dynamic behaviour. Bulletin of Earthquake Engineering. Epub ahead of print 4 August 2018. DOI: 10.1007/s10518-018-0432-7

19.

Valluzzi

Garbin

Modena

et al . (2016) Modeling of timber floors strengthened with seismic improvement techniques. Journal of Heritage Conservation 46: 69–79.

20.

Wilson

Quenneville

PJH

Ingham

(2014) In-plane orthotropic behavior of timber floor diaphragms in unreinforced masonry buildings. Journal of Structural Engineering 140(1): 04013038.

21.

Dolan

(2009) Development of a wood-frame shear wall model in ABAQUS. Journal of Structural Engineering 135(8): 977–984.