Evaluation of the Ultimate Lateral Strength of SWP
1.1. Lateral Strength of SWP
The lateral strength of SWP is primarily contributed by the sheathing and framing studs and can expressed as,
Pr = Psf + Pf Eq. ( 1)
where PSf and PF are lateral strengths associated with the sheathing and framing studs, respectively. In the case that the sheathing are provided in both sides of SWP, the lateral strength of the sheathing is given by
PSf = PSf,1 + PSf,2 Eq. ( 2)
where Psf,1 and Psf,2 are the lateral strength of sheathing presented on side 1 and 2 of the panel, respectively. In addition to the material and cross-section properties of sheathing, the ultimate lateral strength of sheathing are also highly affected by the characteristics and arrangement of sheathing-to-framing connections, which will be discussed in Section 2.2. The lateral strength of framing studs, PF, can be determined as
Pf = Kf A Eq. ( 3)
where Kf is the lateral stiffness of the framing studs and A is the lateral deflection the SWP impending the failure. Compared to that of the sheathing, the framing studs contribute little to the ultimate lateral strength of SWP, as the lateral stiffness of the studs is insignificant. Therefore, for the reason of simplicity, the elastic lateral stiffness of the framing studs is adopted as
KF = І Eq. ( 4)
where Ef and IF are the Young’s modulus and the moment of inertia of the framing studs, respectively. h is the height of the panel.
Considering the compatibility of lateral deformation between sheathing and framing studs prior to the failure of the panel, the relationship between the sheathing strength and the lateral deformation of the panel is,
1.2. Lateral Stiffness and Strength of Sheathing
where Es and GS are the Young’s and shear modulus of sheathing, respectively; h is the height of the panel; and AC is the reduced cross sectional area of the sheathing, defined as
AC = tSd CnC Eq. ( 9)
in which ts is the thickness of the sheathing; dC is the diameter of the screws, nC is the number of screws along the cross section of the sheathing that is connected to the top collector member, and IS is the moment of inertia of the reduced cross-section and is given by
Is = nC tjSfT + 2tSdC Z (i ■ sC )2 Eq. ( 10)
where sc is the screw spacing at the edge of the panel.
Considering the analogy between SWP and the eccentrically loaded bolted connection, in both cases the loads are applied eccentrically and the strength reductions are as a result of the failures of the connections or fasteners initiated at locations which are far from the centre of rotation. In this research, the inelastic method of evaluating strength of the eccentrically loaded bolted connection proposed by Brandt (1982) is employed and extended to evaluate the ultimate lateral strength of sheathing. Brandt’s method involved an iterative process of locating the inelastic
instantaneous center of rotation of the bolt group as shown in Figure 1; the ultimate strength of the connection is found when all of the forces (both internal and external) on the connection are in equilibrium. Extended from Brandt’s method, the ultimate lateral strength of sheathing, PSfi (z’=1, 2) can be evaluated as
PSf, k = C( Vr (k = 1, 2) Eq. (11)
where nx is the index of the last iteration, and Vr is the strength of a single sheathing-to-framing connection which is determined by the minimum value of the bearing resistance of the sheathing material, the shear resistance of the fastener, and the bearing resistance of the steel stud. The parameter C() is the ultimate strength reduction coefficient for the group of sheathing-to – framing connections and can be evaluated through the following procedure.
The coordinates of the inelastic instantaneous center shown in Figure 1 are given by
The initial load eccentricities with respect to the inelastic instantaneous center are
where є is a pre-assigned tolerance for convergence. The iterative process is terminated when the coefficient Cu will be invariant in further iterations, which indicates the equilibrium conditions are satisfied with respect to the updated location of inelastic instantaneous centre. As stated by Brandt (1982), and also found by this study, only a few iterations are required to obtain the ultimate strength reduction coefficient.
Having computed the ultimate strength reduction coefficient, the strength of the sheathing can be calculated based on Eq. (11) and the ultimate lateral strength of SWP can then be determined in accordance with Eq. (7).