Theoretical analysis
In the steel–bamboo SI system, the stability of the steel frame is enhanced because the bamboo infilled wall acts as diagonal bracing. Because the low cyclic loading is used in this test, the infilled wall is needed in order to play a supporting role in both push and pull directions, so the equivalent cross bracing is more reasonable. Based on the above principle, the lateral stiffness formula of the structure is summarized through theoretical analysis, and the axial stiffness and ultimate bearing capacity of the diagonal bracings are derived.
Lateral stiffness formula
The lateral deformation of the bamboo infilled wall consists of three parts: lateral deformation of the screws on the horizontal keels and the vertical keels, shear deformation of the bamboo infilled wall and bending deformation of the bamboo infilled wall. In order to obtain the formula of lateral stiffness, the following hypotheses are made for bamboo infilled wall:

1.
The horizontal internal force components of horizontal keel will be uniformly distributed along the keel, and the vertical internal force components of vertical keel will be uniformly distributed along the keel.

2.
The influence of vertical internal force component of the horizontal keel and horizontal internal force component of the vertical keel on bamboo infilled wall’s shear deformation is ignored;

3.
The shear stress of the bamboo infilled wall is uniformly distributed in the effective force range;

4.
The initial stiffness and bearing capacity of the screws are equal in all directions;

5.
The vertical deformation of screws of horizontal bamboo keels is neglected, and the lateral deformation of screws of vertical bamboo keels is neglected.
Thus, the overall deformation formula of the wall is Eq. (5):
$$\Delta_{\text{Z}} = \Delta_{\text{b}} + \Delta_{\text{M}} + \Delta_{\text{Q}}$$
(5)
where \(\Delta_{\text{Z}}\) is the total deformation of bamboo infilled wall; \(\Delta_{\text{b}}\) is the lateral deformation of the bamboo infilled wall caused by the screws; \(\Delta_{\text{M}}\) is the bending deformation; and \(\Delta_{\text{Q}}\) is the shear deformation.
The total deformation of the bamboo infilled wall can be given in Eq. (6):
$$\Delta_{\text{Z}} = \frac{{F_{\text{Z}} }}{{K_{\text{Z}} }}$$
(6)
where \(F_{\text{Z}}\) is the lateral force shared by bamboo infilled wall and \(K_{\text{Z}}\) is the lateral stiffness of bamboo infilled wall.
The bolts in this test have high strength and the lateral displacement of the bolts is extremely small, so only screws are considered in this paper. The deformation of screws is divided into two parts: the lateral deformation of screws of the horizontal keels and the vertical deformation of screws of the vertical keels. Respective screws have different deformations by bending deformation of bamboo wall, but in this paper the average deformation value is used to simplify the formula. So, the lateral deformation of the bamboo infilled wall caused by the screws can be expressed as Eq. (7):
$$\Delta_{\text{b}} = \frac{{2F_{\text{Z}} }}{{n_{\text{b}} K_{\text{b}} }} + 2 \cdot \frac{{F_{\text{Z}} \cdot {{h_{\text{w}} } \mathord{\left/ {\vphantom {{h_{\text{w}} } {b_{\text{w}} }}} \right. \kern0pt} {b_{\text{w}} }}}}{{n_{\text{h}} K_{\text{b}} }} \cdot \frac{{b_{\text{w}} }}{{h_{\text{w}} }} = \frac{{2F_{\text{Z}} }}{{K_{\text{b}} }} \cdot \left( {\frac{1}{{n_{\text{b}} }} + \frac{1}{{n_{\text{h}} }}} \right)$$
(7)
where \(n_{\text{b}}\) is the number of screws on the horizontal keels; \(n_{\text{h}}\) is the number of the screws on the vertical keel; and \(K_{\text{b}}\) is the stiffness of twin shear screw node.
The shear deformation of bamboo infilled wall is shown in Eq. (8):
$$\Delta_{\text{Q}} = \frac{{F_{\text{Z}} }}{{G_{\text{Z}} tb_{\text{w}} }} \cdot h_{\text{w}}$$
(8)
where \(G_{\text{Z}}\) is the transverse shear modulus of the wall; \(t\) is the thickness of the wall; \(b_{\text{w}}\) is the effective width of the wall which is surrounded by screws; and \(h_{\text{w}}\) is the effective height of the wall which is surrounded by screws.
The inflection point of the wall is located at the center section; thus, the bending deformation of the bamboo infilled wall can be regarded as the sum of the upper and lower part deformations. Each part is assumed to be a cantilever, and the deformation can be given in Eq. (9):
$$\Delta_{\text{M}} = 2 \cdot \frac{{F_{\text{Z}} \cdot ({{h_{\text{w}} } \mathord{\left/ {\vphantom {{h_{\text{w}} } {2)^{3} }}} \right. \kern0pt} {2)^{3} }}}}{{3E_{\text{Z}} I_{\text{W}} }} = \frac{{F_{\text{Z}} h_{\text{w}}^{3} }}{{12E_{\text{Z}} \cdot \frac{{tb_{\text{w}}^{3} }}{12}}} = \frac{{F_{\text{Z}} h_{\text{w}}^{3} }}{{E_{\text{Z}} tb_{\text{w}}^{3} }}$$
(9)
where \(E_{\text{Z}}\) is the bending elastic modulus of the wall and \(I_{\text{w}}\) is the effective section moment of inertia.
To summarize, lateral stiffness of bamboo infilled wall can be expressed as Eq. (10):
$$K_{\text{Z}} = \frac{1}{{\frac{2}{{n_{\text{b}} K_{\text{b}} }} + \frac{2}{{n_{\text{h}} K_{\text{b}} }} + \frac{{h_{\text{w}} }}{{G_{\text{Z}} tb_{\text{w}} }} + \frac{{h_{\text{w}}^{3} }}{{E_{\text{Z}} tb_{\text{w}}^{3} }}}}$$
(10)
The lateral stiffness of steel frames \(K_{\text{s}}\) is Eq. (11):
$$K_{\text{S}} = \frac{{24E_{\text{S}} I_{\text{S}} }}{{h_{\text{G}}^{3} }}$$
(11)
where \(E_{\text{s}}\) is the elastic modulus of the steel column; \(I_{\text{s}}\) is the moment of inertia of the steel columns; and \(h_{\text{G}}\) is the story height of the steel frame.
Thus, the overall lateral stiffness of the steel–bamboo SI system can be given in Eq. (12):
$$K_{\text{Z}} = \frac{1}{{\frac{2}{{n_{\text{b}} K_{\text{b}} }} + \frac{2}{{n_{\text{h}} K_{\text{b}} }} + \frac{{h_{\text{w}} }}{{G_{\text{Z}} tb_{\text{w}} }} + \frac{{h_{\text{w}}^{3} }}{{E_{\text{Z}} tb_{\text{w}}^{3} }}}} + \frac{{24E_{\text{S}} I_{\text{S}} }}{{h_{\text{G}}^{3} }}$$
(12)
Equivalent diagonal bracing formula
The equivalent bracing of bamboo infilled wall is shown in Fig. 13, and the stability of diagonal braces is ignored. There are two main factors limiting the bearing capacity of bamboo infilled wall: one is the bearing capacity of screws, and another is the strength of bamboo infilled wall. It can be observed from the lowcycle reversed loading test that the bamboo wall is not damaged when the screws reach the yield load. Therefore, the yield load of the screws should be taken as the design load capacity (i.e., the 50% of the ultimate load of the screw). It is written in Eq. (13):
$$P_{\text{a}} \le P_{\text{ay}} ,\;\;\Delta_{\text{a}} \le \Delta_{\text{ay}}$$
(13)
where \(P_{\text{a}}\) is the internal forces for screws and partial safety can be taken as \(P_{\text{a}} = \sqrt {P_{\text{ah}}^{2} + P_{\text{av}}^{2} }\), in which, \(P_{\text{ah}} = F_{\text{Z}} /n_{\text{b}}\) is the average internal force of the screws on the horizontal bamboo keel, \(P_{\text{av}} = F_{\text{Z}} h_{\text{w}} /n_{\text{h}} b_{\text{w}}\) is the average internal force of the screws on the vertical bamboo keel; \(P_{\text{ay}}\) is the yield bearing capacity for screws; \(\Delta_{\text{a}}\) is the corresponding deformation of \(P_{\text{a}}\); and \(\Delta_{\text{ay}}\) is the corresponding deformation of \(P_{\text{ay}}\).
The parameters in Eq. (13) are brought to get the allowable bearing capacity and deformation of the bamboo infilled wall. It can be shown in Eq. (14):
$$F_{\text{Z}} \le \frac{{P_{\text{ay}} n_{\text{b}} n_{\text{h}} b_{\text{w}} }}{{\sqrt {n_{\text{b}}^{2} h_{\text{w}}^{2} + n_{\text{h}}^{2} b_{\text{w}}^{2} } }},\;\;\;\Delta_{\text{Z}} \le \frac{{K_{\text{b}} \Delta_{\text{ay}} n_{\text{b}} n_{\text{h}} b_{\text{w}} }}{{K_{\text{Z}} \cdot \sqrt {n_{\text{b}}^{2} h_{\text{w}}^{2} + n_{\text{h}}^{2} b_{\text{w}}^{2} } }}$$
(14)
The overall deformation of the steel frame–bamboo infilled wall can be expressed in Eq. (15):
$$\Delta_{\text{Z}} ' = \Delta_{\text{Z}} + \frac{{h_{\text{G}}^{3} }}{{24E_{\text{S}} I_{\text{S}} }}$$
(15)
The height of the steel frame is \(h_{\text{G}}\), the span of the steel frame is \(b_{\text{G}}\), the length of the equivalent diagonal bracing is L and the angle between the bracing and the horizontal direction is θ. According to the geometric relationship, there is:
$$L = \sqrt {b_{\text{G}}^{2} + h_{\text{G}}^{2} }$$
(16)
Using \(L\) as a function of \(b_{\text{G}}\) and deriving from \(b_{\text{G}}\), Eq. (17) can be obtained.
$${\text{d}}L = {\text{d}}b_{\text{G}} \cdot \frac{{b_{\text{G}} }}{L} = {\text{d}}b_{\text{G}} \cdot \cos \theta$$
(17)
Then,
$${\text{d}}L = \Delta_{\text{z}}^{{\prime }} \cdot \cos \theta$$
(18)
Therefore, the \(\Delta_{\text{z}}^{{\prime }}\) can be expressed by the axial deformation of the bracing as Eq. (19):
$$\Delta_{\text{Z}} ' = \frac{{F_{\text{Z}} (b_{\text{G}}^{2} + h_{\text{G}}^{2} )^{{\frac{3}{2}}} }}{{2E_{\text{x}} A_{\text{x}} b_{\text{G}}^{2} }}$$
(19)
where \(E_{\text{x}} A_{\text{x}}\) is the axial stiffness of the equivalent diagonal bracing.
Equations (6), (15) and (19) can be used to obtain the expression of axial stiffness of equivalent diagonal bracing:
$$E_{\text{x}} A_{\text{x}} = \frac{{12E_{\text{S}} I_{\text{S}} F_{\text{Z}} \cdot (b_{\text{G}}^{2} + h_{\text{G}}^{2} )^{{\frac{3}{2}}} }}{{b_{\text{G}}^{3} h_{\text{G}}^{3} + 24b_{\text{G}}^{2} \Delta_{\text{Z}} E_{\text{S}} I_{\text{S}} }}$$
(20)
The axial force of each diagonal bracing should satisfy Eq. (21):
$$F_{\text{x}} \le \frac{1}{2} \cdot \frac{{\sqrt {b_{\text{G}}^{2} + h_{\text{G}}^{2} } }}{{b_{\text{G}} }} \cdot \frac{{P_{\text{ay}} n_{\text{b}} n_{\text{h}} b_{\text{w}} }}{{\sqrt {n_{\text{b}}^{2} h_{\text{w}}^{2} + n_{\text{h}}^{2} b_{\text{w}}^{2} } }}$$
(21)
The formulas mentioned above are mainly aimed at structural design, and no specific study has been made on the overall yield of the system. Therefore, only the equivalence of the system in the linear elastic stage is emphasized.
Comparison of theory and experiment
In order to verify the correctness of the simplified formula, the elastic lateral stiffness obtained by theoretical calculation is compared with the elastic stiffness obtained by the experiment. The material properties of bamboo scrimber and screws are referred from Table 1.
The elastic stiffness obtained from the experiment is \(K^{\prime} = 1 9. 9 6\;\;{\text{kN/mm}}\) as shown in Table 2. And according to Eq. (12), the elastic stiffness is calculated as follows:
$$K_{\text{Z}} = \frac{1}{{\frac{2}{{n_{\text{b}} K_{\text{b}} }} + \frac{2}{{n_{\text{h}} K_{\text{b}} }} + \frac{{h_{\text{w}} }}{{G_{\text{Z}} tb_{\text{w}} }} + \frac{{h_{\text{w}}^{3} }}{{E_{\text{Z}} tb_{\text{w}}^{3} }}}} + \frac{{24E_{\text{S}} I_{\text{S}} }}{{h_{\text{G}}^{3} }} = 20.23{\text{ [kN/mm]}}$$
Comparing the test results with the theoretical calculation results, the theoretical calculation value is greater than the test value, and the relative error between the two is about 1.3%. The main reasons for this result are: In the actual test, the internal forces of screws and infilled walls are not uniform; the stress concentration phenomenon occurred in the screw joint; the actual bamboo infilled walls are anisotropic materials; and the mechanical properties of the transverse and longitudinal walls are quite different. However, these factors have little effect on the lateral displacement of bamboo infilled wall, and the test value is in good agreement with the theoretical calculation value. Based on the above considerations, the lateral stiffness formula proposed in this paper can be basically used to calculate the lateral stiffness of infilled walls.