Fourth International Conference on FRP Composites in Civil Engineering (CICE8) 22-24July 8, Zurich, Switzerland Dynamic and buckling analysis of FRP portal frames using a locking-free finite element F. Minghini, N. Tullini & F. Laudiero Department of Engineering, University of Ferrara, Ferrara, Italy ABSTRACT: Vibration frequencies in the presence of geometrical effects, and buckling loads of FRP pultruded portal frames were determined for different restraint and load conditions. The kinematical model adopted is based on a second-order approximation of the displacement field and accounts for shear strain effects due to non-uniform bending and torsion. The seven unknown displacement functions were interpolated by a set of locking-free modified Hermitian polynomials. Buckling curves for one- and three-bay frames were determined, showing the influence of shear deformations and the stiffening effect of warping constraints. The role played by participant masses and external load second-order effects in dynamic response was pointed out. Frequency crossing between in-plane and out-of-plane fundamental vibration modes was observed. 1 INTRODUCTION Thin-walled pultruded profiles made of fiber-reinforced polymers (FRP) have increasingly been used in many engineering applications when characteristics of lightness and durability are of primary importance. As a matter of fact, performances of FRP profiles are basically ruled by deformability and buckling phenomena. In particular, the high ratio between longitudinal and transverse elastic moduli typically implies a strong influence of shear deformations (Cortínez & Piovan 2). In recent research works concerning pultruded frames, the stiffening effect of warping restraints at the column bases was put in evidence (Minghini et al. 8a). Moreover, it was shown that, in the presence of a low self weight, participant additional masses strongly influence the dynamic response (Minghini et al. 8b). In this paper, a numerical beam model is adopted for dynamic and buckling analysis of FRP pultruded portal frames. The displacement field assumed is based on a second-order approximation of the finite rotation tensor (Chang et al. 1996), taking the shear strain effects due to nonuniform bending and torsion into account. The equation of motion was determined via Hamilton s principle. Rotatory inertia terms are included in the kinetic energy expression. The elastic strain energy functional contains coupling terms between shear resultants and non-uniform torsion. Hence, shear-deformable beams with generic cross-sections can be analyzed. Moreover, the adopted expression for the second-order work of external forces holds for generic surface loads and accounts for possible load eccentricities with respect to the shear centre. Stiffness and mass matrices were consistently derived interpolating the seven unknown displacement functions by means of locking-free shape functions of the Hermitian family (Minghini et al. 7), containing the parameters 12EJ y 12EJ x φ x = ; 2 y 2 GD l φ = GD l ; 12EJω φ ω = (1) GD l 2 x y ω - 1 -
In Equation 1, EJ x, EJ y and EJ ω are the flexural and warping rigidities, whereas GD x, GD y and GD ω represent the rigidities associated to shear resultants and non-uniform torsion. Finally, l stands for the element length. For shear-undeformable beams, φ x = φ y = φ ω = and the classical Euler-Bernoulli-Vlasov element is re-obtained. 2 BUCKLING ANALYSIS 2.1 Example 1. GFRP portal frame under in-plane horizontal nodal forces The portal frame shown in Figure 1a was analyzed for the beam length l ranging from 1.5 m to 4.5 m. All members, showing E = GPa and G = E/1, are GFRP-pultruded I-section profiles having the major axis (x loc ) parallel to the global y-axis. Full displacement and rotation continuity, and no warping restraint were assumed at the column-beam joints. To assure an accurate result (Minghini et al. 8a) four beam finite elements were adopted for each member. In order to highlight the bracing influence on the frame response, three constraint conditions were analyzed: lateral displacements of the entire columns, of the entire beam, and of nodes B and C only, were alternatively restrained. The dashed curve of Figure 2a gives the buckling load when column lateral displacements are prevented. In the corresponding buckling mode (Fig. 1b), the beam shows antisymmetrical lateral displacements and symmetrical torsional rotations, whereas the columns present antisymmetrical torsional rotations with respect to each other. The two dash-dot curves of Figure 2a refer to laterally restrained beam displacements. In particular, the upper curve corresponds to the case of restrained warping at the column bases, whereas the lower curve corresponds to the case of free warping. It should be noted that, as the beam length increases, the axial compressive force in column C-D lowers so as to increase the frame buckling load. In the corresponding buckling mode (Fig. 1c), the compressed column shows lateral deflections coupled with torsion. In particular, restraining warping at the column bases implies a strong increase (+4%) of the buckling load. Another significant solution is obtained when the out-of-plane displacements of joints B and C are prevented. In this case (thick solid lines of Fig. 2a), the buckling curves border the two limit cases described above, reaching a maximum for l h. Finally, the two thin solid curves of Figure 2a refer to the case of free lateral displacements, assuming warping restraints at the column bases or not. In both cases, for relatively short beams, the (antisymmetrical) critical shape (Fig. 1d) is characterized by the flexural-torsional column displacements and consequently influenced by the presence of base warping restraints. Nevertheless, as the beam length increases, both curves tend to reduce to the case of laterally restrained columns. The two cases of frames with no lateral restraints or with restraints at beamcolumn nodes only (with column bases free to warp) were reanalyzed by developing a 3-D plate-shell FE model (STRAUS7 4). The 3-D model was able to capture local instability at the column bases and in the proximity of the beam ends. Hence, the buckling load decreased (not more than 4%) with respect to the corresponding beam solution. a b c d Figure 1. a: FRP pultruded portal frame subjected to two equal horizontal forces at the beam-column nodes. Example 1: h = 3. m, l = 1.5 4.5 m, x loc // y, H = 24. mm, B = 1. mm, t w = t f = 12. mm. Example 2: h = 2. m, l = 1.5 4.5 m, x loc // y, H = B = 11.6 mm, t w = t f = 6.5 mm. Critical mode shapes for laterally restrained columns (b), laterally restrained beam (c) and no lateral restraint (d). Straight and circular arrows indicate lateral displacements (y direction) and torsional rotations respectively. - 2 -
4 35 base warping prevented 1 9 E /G = 35 (CFRP) Fx,cr (kn) shell solutions base warping free Fx,cr (kn) 6 1.5 2. 2.5 3. 3.5 4. 4.5 l (m) a shear strength limit E /G = 1 (GFRP) 1.5 2. 2.5 3. 3.5 4. 4.5 l (m) b Figure 2. Bucking load of the frame of Fig. 1a for increasing beam length. a: I-section profiles with different restraint conditions: laterally restrained columns ( ), laterally restrained beam ( ), lateral restraints at nodes B and C (, ), no lateral restraint (, ). b: wide-flange GFRP or CFRP profiles with lateral restraints at nodes B and C, considering (, ) or neglecting (, ) shear deformations. Horizontal dotted line corresponds to shear strength of GFRP columns ( MPa). 2.2 Example 2. Influence of shear deformations The portal frame of Figure 1a (h = 2. m) was re-analyzed to emphasize the role of shear deformations. To this purpose, the critical load was evaluated in both cases (Fig. 2b) of Glass-FRP profiles (E = GPa, G = E/1) and Carbon-FRP profiles (E = 148 GPa, G = E/35). In particular, wide-flange cross-section beams were considered. Out-of-plane bracings were placed at the beam-column joints only and warping restraints were introduced at the column bases and at the beam end sections. Figure 2b shows that, for l 2.5 m, neglecting shear deformations overestimates the buckling load by about 1% for GFRP profiles and by about 35% for CFRP profiles. 2.3 Example 3. Three-bay frame subjected to a uniformly distributed (wind type) lateral load and to a vertical load uniformly distributed along the beam A three-bay, cable-carrying, pultruded frame was then analyzed, showing the influence of possible diagonal stiffeners (Fig. 3). The beam has a C-shaped cross-section, whereas columns are wide-flange profiles. Both columns and beam have the weak axis (y loc ) parallel to global y-axis. Columns are fully clamped at the base and free to warp at both ends. At nodes E, F, G and H, full continuity of rotations and displacements is imposed between beams and columns. The outof-plane displacements are prevented at joints E and H only. A uniform lateral load acting on the whole frame was considered. Moreover, a vertical load was uniformly applied along the beam EH. Figure 3. Three-bay cable-carrying GFRP frame (h = l 1 = 2.5 m, l 2 = 4. m) subjected to a uniformly distributed lateral load and to a vertical load acting along the beam. Beam cross-section: B b = mm, H b = 8 mm, t b = 1 mm. Column cross-section: B c = 3.2 mm, H c = 3.2 mm, t c = 9.5 mm. Stiffener crosssection: hollow square tubes 11.6 11.6 6.4 mm. - 3 -
q z,cr (kn/m) 5 45 4 35 1 5 flexural strength limit with stiffeners with no stiffeners flexural strength limit 2 4 6 8 q y,cr (kn/m) Figure 4. Critical-load interaction-curves for the pultruded frame of Figure 3. Horizontal dashed lines correspond to the beam flexural strength ( MPa). The frame buckling was analyzed in the presence of diagonal stiffeners or not. Four finite elements were adopted for beams EF and GH, and for columns, whereas six elements were used for beam FG. The stiffeners, having hollow square section, were modelled using truss-type elements. The critical load interaction curves are summarized in Figure 4. For predominant lateral loads, buckling is governed by beam flexural-torsional instability, i.e.: beam deflection into the vertical (weak) plane coupled with torsion. In turn, due to the presence of one single symmetry axis, torsion is necessarily coupled with flexure into the plane of major stiffness (x-y). Viceversa, when the vertical load prevails, columns BF and CG tend to buckle out of the frame plane taking the beam along. Figure 4 shows that the stiffeners significantly increase the buckling load in the case of predominant lateral forces only. In the presence of vertical loads, the beam can reach the flexural strength limit before buckling takes place (horizontal lines of Fig. 4). 3 VIBRATION ANALYSIS WITH GEOMETRICAL NONLINEARITIES 3.1 Example 4. GFRP portal frame under in-plane distributed vertical loads A pultruded GFRP portal frame is reported in Figure 5a (E = GPa, G = E/1, ρ = 1.83 t/m 3 ). All members are wide-flange profiles and have the major axis (x loc ) aligned with the global y- axis. Full displacement and rotation continuity was assumed at the column-beam joints, whereas top and bottom column end-sections were considered free to warp. Furthermore, nodes B and C were assumed to be laterally supported. A uniformly distributed gravitational load was applied at the beam top flange. a b c Figure 5. a: GFRP portal frame subjected to a uniform gravitational load acting at the beam top flange; E = GPa, G = E/1, ρ = 1.83 t/m 3, H = B = 3.2 mm, t w = t f = 9.5 mm, x loc // y. Example 4: l = h = 3. m (Fig. 6a) or l = 3. m, h = 2. m (Fig. 6b). Example 5: l = 4. m, h = 3.5 m. Fig. b: in-plane (flexural) vibration mode. Fig. c: out-of-plane (flexural-torsional) vibration mode. - 4 -
1st mode 2nd mode α = 1st mode 2nd mode f (Hz) 1 5 α = α =.2 α = 1. 5 1 35 q z (kn/m) a f (Hz) 1 5 α =.2 α = 1. 5 1 35 q z (kn/m) b Figure 6. Portal frames with l = h = 3. m (a) and l = 3. m, h = 2. m (b). First and second natural frequencies vs. gravitational load uniformly distributed along the beam. The mass participating in the frame vibrations is alternatively assumed to be %, % and 1% of the mass associated to the (static) live load. Symbol indicates frequency crossing. 2. f (Hz) 1.5 1. 1st mode 2nd mode shear strength limit.5. 5 1 35 q z (kn/m) Figure 7. Portal frame with l = 4. m, h = 3.5 m (α = 1). First and second natural frequencies, including shear deformations (, ) or not (, ), vs. gravitational load uniformly distributed along the beam. Vertical dashed line corresponds to the beam shear strength ( MPa). For analysis purposes, the corresponding mass per unit length was given the expression (α q z )/g and was assumed fully (α = 1) or partly (α =.2) participant, or totally ineffective (α = ). Geometrical effects induced by applied loads were finally considered and an accurate solution (Minghini et al. 8b) was searched for, adopting four finite elements for each member. Assuming l = h = 3. m, the case without participant masses (α = ) was examined first. In this case (Fig. 6a), the first natural frequency substantially remains constant for a long load range, revealing a negligible influence of second-order effects. Then, for q z /q z,cr.75, a frequency crossing is observed (symbol ). In fact, for q z.75 q z,cr, the first modal shape is characterized by in-plane vibrations (Fig. 5b), whereas, for q z.75 q z,cr, out-of-plane (flexuraltorsional) vibrations occur (Fig. 5c). Compared with self weight, even small participant masses significantly reduce the natural frequencies. For example, for q z = 3 kn/m, the first frequency decreases of about 6% when α ranges from to.2. For higher l/h ratios, the load corresponding to the frequency crossing drastically decreases. For example, assuming l/h = 3./2., the crossing is observed at q z /q z,cr.3 for α =.2 (Fig. 6b). Moreover, it could be shown that, if second-order effects are neglected, the first frequency is overestimated of about % for q z /q z,cr =.75. - 5 -
3.2 Example 5. Influence of shear deformations The portal frame of Figure 5a (l = 4. m, h = 3.5 m) was finally reconsidered to enlighten the role of shear deformations for the limit case α = 1. In fact, Figure 7 shows that ignoring shear deformations implies an overestimate of the first frequency of about 1% before the frequency crossing occurs (Fig. 7). Furthermore, if shear deformations are neglected, the load at frequency crossing, as well as the buckling load, turn out to be overestimated of about 13%. 4 CONCLUSIONS Buckling loads and vibration frequencies of FRP pultruded portal frames were determined using a locking-free finite element formulation. In particular, it was shown that: the model proposed agrees well with 3D FE solutions (differences not larger than 4%) over a wide range of frame span/height ratios; shear deformations may give rise to appreciable reductions of buckling loads and vibration frequencies (1% 13% and ~35% in the examples concerning GFRP and CFRP profiles, respectively); restraining column base warping may imply a strong increase of the buckling load (+4% for the case examined); masses of the live loads (participating in the vibration response) remarkably reduce the natural frequencies and may cause sudden exchanges between in-plane and out-of-plane vibration modes; geometrical effects due to external loads may significantly reduce the fundamental frequency for sufficiently high ratios span/height. The stiffness and mass matrices adopted could be usefully implemented in commercial codes for linear stability and frequency analysis of spatial beam assemblages. 5 ACKNOWLEDGMENTS The present investigation was developed in the framework of two coordinated Projects: the Italian Research Program no. 589782 coordinated by Prof. Franco Maceri from University of Rome ''Tor Vergata'' and the Research Program FAR 7 of the University of Ferrara. Financial support of the Italian Ministry of University and Research, and of the University of Ferrara is gratefully acknowledged. REFERENCES Chang, S.-P., Kim, S.-B., and Kim, M.-Y. 1996. Stability of shear deformable thin-walled space frames and circular arches. J Eng Mech ASCE, 122, 844-854. Cortínez, V. H., and Piovan, M. T. 2. Vibration and buckling of composite thin-walled beams with shear deformability. J Sound Vibr, 8, 71-723. Minghini, F., Tullini, N., and Laudiero, F. 7. Locking-free finite elements for shear deformable orthotropic thin-walled beams. Int J Numer Meth Eng, 72, 88-834. Minghini, F., Tullini, N., and Laudiero, F. 8a. Buckling analysis of FRP pultruded frames using locking-free finite-elements. Thin-Wall Struct, 46: 223-241. Minghini, F., Tullini, N., and Laudiero, F. 8b. Vibration analysis with second-order effects of FRP pultruded frames using locking-free elements. Submitted for publication. STRAUS7. 4. Theoretical manual theoretical background to the Straus7 finite element analysis system, first edition. - 6 -