1614 Design of Pultruded Structural Members

Fundamentals needed for the design of transversely loaded members (beams), axially loaded compression members (columns), and axially loaded tension members (ties, truss members) are presented in the following. Connections for beams and columns are treated in the Section 16.1.5. The limit states discussed here for the members assume that the connections are appropriately designed to transfer all loads into the members. For each type of member various limit states are identified and discussed.

Analytically derived equations are presented for analyzing strength, stability, and deformation limit states. While the equations may be appropriately used with any of the four design basis approaches discussed previously, they are only valid if appropriate factors can be determined for the conditions for which they are used. The equations are primarily those presented in the SPDM (which are in most cases identical to those in the EDCH) and can therefore reasonably be assumed to be suitable for use in an LSD approach or in an ASD approach. Many of the equations presented may also be found in pultrusion manufacturer design guides, which are a combination of analytical and empirical equations. It can be assumed that the safety factors recommended by pultrusion manufacturers can be used with confidence with the analytical equations presented in this section. In addition, material properties provided by pultrusion manufactures can similarly be used in an ASD approach with the equations provided herein. Transversely Loaded Members

Members considered are symmetric with respect to the flexural plane and consist of horizontal plates (flanges) and vertical plates (webs). Where singly symmetric sections are used it is assumed that they are loaded in the plane of symmetry through the shear center. Typical common pultruded shapes that fall in this definition, and that are routinely used as beams, include I, WF, and tubular ''box'' sections. Channel sections and angles are often used in pairs to form built-up doubly symmetric or singly symmetric sections.

The member is designed to resist stress-resultants that are due to the applied transverse forces (loads). These stress-resultants are the bending moment (M) and the transverse shear force (V). Due to the stress-resultant M the member is assumed to develop an axial (flexural) stress, sz = Mxy (16.7)

Due to the stress-resultant V the member is assumed to develop a transverse shear stress,


Due to local concentrated loads P and reactions R at supports in the plane of the web at the point of load application or at the support, the member is assumed to develop a transverse compressive stress given by

7 Aeff y Aeff where Aeff is the effective area over which the concentrated load or reaction is applied and can be taken as

where n is the number of webs, and Leff is the effective bearing length along the beam and is taken as the width of the support or the length over which the concentrated load is applied. i„, tf, and tbp, are the web thickness, the flange thickness, and the thickness of the bearing plate under the flange (if applicable), respectively.

Due to the stress-resultants M and Vit is assumed that the member will deflect in its plane of loading. The elastic curve that describes the deflected shape is a function of the flexural rigidity of the member (EI) and the transverse shear rigidity of the member (KAG). It can be obtained from the Timoshenko (shear deformation) beam theory as d(z)=M+áM (16.n)

where the functions f1(z) and f2(z) depend on the loading and boundary conditions and can be found in standard engineering texts.

For flexural members SLSs (deflection) and ULSs (strength and stability) are considered. Procedures are presented to determine the short-term and long-term member deflections as a function of the material properties and member geometric properties. The calculated design stresses (or design force resultants) must be less than the critical strengths and critical buckling stresses (the critical resistances or capacities) according to the design basis selected. The calculated deflections (serviceability design deflections) must be less than the building code stipulated deflections. The EDCH (1996) recommends limiting total deflections to the range of L/400 to L/250 for FRP frame structures.

Since pultruded members have low stiffness to strength ratios relative to conventional structural materials, the stiffness controlled limit states are checked first (i.e., deflection and stability). Thereafter, the ultimate strength limit states are checked. The order of the limit states presented follows this order. It may be noted that this is typically not the order in which limit states are considered for design with conventional materials. Transverse Deflection

Shear deformation beam theory is used to determine the deflection of the beam. This is the procedure recommended by the SPDM (1984) and the EDCH (1996). The use of shear deformation beam theory is especially important in FRP pultruded beams due to the relatively low longitudinal modulus (leading to beams with short spans) and the relatively high E/G ratios (and low shear moduli) (see Bank 1989a,b; Mottram 1992). To calculate the design deflection, appropriate choices of the flexural rigidity, EI, and the transverse shear rigidity, KAG, are required. For homogenous (i.e., having the same properties in the flanges and webs of the section) pultruded beams the following are recommended:

where ktim is the (Timoshenko) shear coefficient. It can be found for both homogenous and non-homogenous thin-walled composite beam sections as described by Bank (1987). The flexural rigidity is primarily a function of the longitudinal moduli of the flanges and the webs. For nonhomogenous sections having different longitudinal moduli in the flanges and the webs the conventional mechanics of composite sections or the ''transformed-section'' method can be used to find the effective flexural rigidity of the section. The section transverse shear rigidity is a function of the shear flow in the flanges and the webs of the section (Bank 1987). However, for common pultruded I, WF, and box sections it has been shown that this term can be reasonably replaced by either the full-section shear rigidity defined as AGb or the area of the web multiplied by the in-plane shear modulus of the web, AwGLT (Bank 1989a,b). The full-section shear modulus is found by tests on full-section pultruded beams (Bank 1989a,b; Roberts 2002).

The short-term longitudinal and shear moduli presented in the deflection equation will both decrease as a function of time due to the viscoelastic nature of the FRP composite. To predict the long-term deflection, time-dependent viscoelastic moduli are substituted for the short-term moduli that are typically measured by coupon tests. Both the longitudinal modulus and the in-plane shear modulus will be time dependent; however, their time dependency will be different. In general, the shear modulus will be more time dependent than the longitudinal modulus since it is primarily a resin matrix-dependent property. The viscoeleastic longitudinal modulus, EL = Ev, and the viscoelastic in-plane shear modulus, Glt = Gv, are given as (Bank and Mosallam 1992).

where t is the time in hours, E0 and G0 are the short-term time-independent moduli and Et and Gt are the (constant) time-dependent moduli determined from the sustained loading (or from long-term creep tests), and ne and ng are the empirical constants obtained from curve fitting (from creep tests) according to the linearized version of Findley's theory (SPDM 1984; Bank and Mosallam 1992). For common commercially produced pultruded FRP materials the constants obtained by Mosallam and Bank (1991) based on 2,000 and 10,000-h creep tests are ne = ng = 0.30, E = 180 x 106 psi and Gt = 30 x 106 psi. Coefficients for a variety of plastics and other nonpultruded composites are reported in the SPDM (1984). In typical pultruded structures the long-term sustained stress is between 10 and 20% of the ultimate strength of the material. Global Lateral-Torsional Buckling

Lateral-torsional buckling of symmetric pultruded beams has been studied by a number of researchers. It appears that the well-known equation that is used for isotropic beam sections can be used for standard pultruded I and WF sections provided the appropriate values of E and G are used in the equations (Mottram 1992). Other analytical approaches have been suggested (Barbero and Raftoyiannis 1994; Davalos and Qiao 1997); however, simple design equations are not presented in these works. Inclusion of the effects of shear deformation is discussed by Roberts (2002). The equation to determine the critical lateral-torsional buckling stress for an I or WF section loaded through its centroid (not including the effects of shear deformation) is

C1 /P2ELIyGLT/ P4E2IyIm . .

where C1 is a coefficient that accounts for moment variation along the beam, Sx = Ix/c is the section modulus about the strong (bending) axis, J is the torsional constant, Iy is the second moment about the weak (vertical) axis, Io is the warping constant, kf is an end restraint coefficient for flexural buckling, ko is an end restraint coefficient for torsional buckling, and L is the unbraced length of the member. For WF sections Io = Iy(d2/4), where d is the section depth.

For closed cross-sections such as rectangular tubes the warping torsional resistance is large and lateral buckling is not a critical condition. For singly symmetric sections such as channels and angles little experimental data is available for pultruded members. Use of appropriate equations for isotropic metallic members is recommended at this time with substitution of the isotropic material properties with those of the orthotropic material properties. Local Compressive Flange Buckling

Transversely loaded thin-walled sections can fail due to local buckling of the compressive flange (Barbero et al. 1991; Bank et al. 1994b, 1995). The critical buckling load (or stress) in the flange is a function of the boundary conditions on the longitudinal edges of the flange. In I and WF sections one edge is free, while the other edge is elastically restrained. There is no exact closed-form solution for this case. For the characteristic equations and computations see Bank and Yin (1996). For I and WF beams both the SPDM (1984) and the EDCH (1996) recommend that the elastically restrained edge be assumed to be simply supported. This is a conservative assumption. Test results show clearly that the local buckling stress is higher than that predicted by this assumption; however, it is lower than that predicted by assuming that the edge is fixed (see Bank et al. 1995). The Equations presented below are taken from the SPDM (1984) and are attributed to Haaijer (1957). They are shown in terms of the effective engineering properties of the section in terms of the flexural stiffness coefficients (Dj) in the SPDM.

Assuming the restrained edge to be simply supported, the local buckling stress for flanges in compression is calculated as follows:


where tf is the flange thickness, bf is the (entire) flange width, and a is the unbraced length of the flange. For long unbraced flanges the first term in the square brackets is negligible and the critical stress can be found from scr = 4Glt ( f

If the restrained edge is assumed to be fixed (i.e., built-in or clamped), then the local buckling stress is calculated as follows:

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