Effect of bending test setup on the structural stability of I-shape pultruded FRP profiles using variable step size FE analysis

Figures & data

Figure 1. Typical bending test setups: (a) three-point bending and (b) four-point bending.

Table 1. Failure modes and associated damage formulae utilised in the Hashin damage model Simulia, (2019)

Failure Mode	Damage Initiation^a	Damage Evolution^b
Fibre tension ($\mathit{\sigma }{\text{^}}_{11}$ ≥ 0)	${F_f}^{\mathit{t}}={\left(\frac{\mathit{\sigma }{\text{^}}_{11}}{X^T}\right)}^2+\mathit{\alpha }{\left(\frac{\mathit{\tau }{\text{^}}_{12}}{S^L}\right)}^2$	${{\mathit{\delta }}^{\mathit{ft}}}_{\mathit{eq}}=L^c\sqrt{{⟨{\mathit{\epsilon }}_{11}⟩}^2+\mathit{\alpha }{\left({\mathit{\epsilon }}_{12}\right)}^2},$ ${{\mathit{\sigma }}^{\mathit{ft}}}_{\mathit{eq}}=\frac{{\mathit{\sigma }}_{11}{\epsilon }_{11}+\mathit{\alpha }{\tau }_{12}{\epsilon }_{12}}{{{\mathit{\delta }}^{\mathit{ft}}}_{\mathit{eq}}/L^c}$
Fibre compression ($\mathit{\sigma }{\text{^}}_{11}$ < 0)	${F_f}^{\mathit{c}}={\left(\frac{\mathit{\sigma }{\text{^}}_{11}}{X^c}\right)}^2$	${{\mathit{\delta }}^{\mathit{fc}}}_{\mathit{eq}}=L^c⟨-{\epsilon }_{11}⟩,$ ${{\mathit{\sigma }}^{\mathit{fc}}}_{\mathit{eq}}=\frac{⟨-{\mathit{\sigma }}_{11}⟩⟨-{\epsilon }_{11}⟩}{{{\mathit{\delta }}^{\mathit{fc}}}_{\mathit{eq}}/L^c}$
Matrix tension ($\mathit{\sigma }{\text{^}}_{22}$ ≥ 0)	${F_m}^{\mathit{t}}={\left(\frac{\mathit{\sigma }{\text{^}}_{22}}{Y^T}\right)}^2+{\left(\frac{\mathit{\tau }{\text{^}}_{12}}{S^L}\right)}^2$	${{\mathit{\delta }}^{\mathit{mt}}}_{\mathit{eq}}=L^c\sqrt{{⟨{\mathit{\epsilon }}_{22}⟩}^2+{\left({\mathit{\epsilon }}_{12}\right)}^2},$ ${{\mathit{\sigma }}^{\mathit{mt}}}_{\mathit{eq}}=\frac{⟨{\mathit{\sigma }}_{22}⟩⟨{\epsilon }_{22}⟩+{\tau }_{12}{\epsilon }_{12}}{{{\mathit{\delta }}^{\mathit{mt}}}_{\mathit{eq}}/L^c}$
Matrix compression ($\mathit{\sigma }{\text{^}}_{22}$ < 0)	${F_m}^{\mathit{c}}={\left(\frac{\mathit{\sigma }{\text{^}}_{22}}{2S^T}\right)}^2+\left({\left(\frac{Y^C}{2S^T}\right)}^2-1\right)\frac{\mathit{\sigma }{\text{^}}_{22}}{Y^C}+{\left(\frac{\mathit{\tau }{\text{^}}_{12}}{S^L}\right)}^2$	${{\mathit{\delta }}^{\mathit{mc}}}_{\mathit{eq}}=L^c\sqrt{{⟨-{\mathit{\epsilon }}_{22}⟩}^2+{\left({\mathit{\epsilon }}_{12}\right)}^2},$ ${{\mathit{\sigma }}^{\mathit{mc}}}_{\mathit{eq}}=\frac{⟨-{\mathit{\sigma }}_{22}⟩⟨-{\epsilon }_{22}⟩+{\tau }_{12}{\epsilon }_{12}}{{{\mathit{\delta }}^{\mathit{mc}}}_{\mathit{eq}}/L^c}$

^a${F_f}^{\mathit{t}}$, ${F_f}^{\mathit{c}}$, ${F_m}^{\mathit{t}}$, and ${F_m}^{\mathit{c}}$ are damage initiation factors for fibre under tension and compression and matrix under tension and compression, respectively. α is the shear stress counter affecting the fibre tensile initiation criterion. $\mathit{\sigma }{\text{^}}_{11}$, $\mathit{\sigma }{\text{^}}_{22}$, and $\mathit{\tau }{\text{^}}_{12}$ refer to the axial, transverse, and shear stress components, respectively, of the effective stress tensor ${\sigma }^{\text{^}}$ responsible for evaluating the initiation criteria. ^b ${{\mathit{\delta }}^{\mathit{ft}}}_{\mathit{eq}}$, ${{\mathit{\sigma }}^{\mathit{ft}}}_{\mathit{eq}}$, ${{\mathit{\delta }}^{\mathit{fc}}}_{\mathit{eq}}$, ${{\mathit{\sigma }}^{\mathit{fc}}}_{\mathit{eq}}$, ${{\mathit{\delta }}^{\mathit{mt}}}_{\mathit{eq}}$, ${{\mathit{\sigma }}^{\mathit{mt}}}_{\mathit{eq}}$, ${{\mathit{\delta }}^{\mathit{mc}}}_{\mathit{eq}}$, and ${{\mathit{\sigma }}^{\mathit{mc}}}_{\mathit{eq}}$ are the equivalent displacement and stress components of the element under fibre tension and compression and matrix tension and compression, respectively. $L^c$ is the characteristic length of the element.

Figure 2. Three shape functions capturing the structural instability of the I-shape pultruded FRP beam.

Figure 3. Typical response of shape function weights versus the analysis time: (a) when lateral-torsional buckling is the dominant failure mode, (b) when local buckling is the dominant failure mode, and (c) when web crippling is the dominant failure mode.

Figure 4. Modelling scheme of variable step size FEM analysis.

Table 2. Dimensions and test setup details of the validation case studies

Specimen ID^a	Cross-sectional dimensions (mm)^b	Beam length (mm)	Type of test setup	Reference
I-LTB	120.0×60.0×6.0	4064	Three-point bending	Nguyen et al. (2014)
I-LB	152.4×101.6×6.35	2900	Four-point bending	Yang et al. (2020)
I-WC	120.0×60.0×6.0	240	Three-point bending	Fernandes et al. (2015)

^aI-LTB refers to an I-shape pultruded FRP beam under lateral-torsional buckling. I-LB denotes an I-shape pultruded FRP beam under local buckling. I-WC points to an I-shape pultruded FRP beam under web crippling. ^b the cross-sectional dimensions of the I-shape beam were labelled as follows: web height × flange width × wall thickness.

Table 3. Characteristic mechanical properties of the I-shape pultruded FRP profiles used in the validation case studies

Reference	Nguyen et al. (2014)	Yang et al. (2020)	Fernandes et al. (2015)
EL (MPa)	23,000	24,500	26,700
ET (MPa)	11,200	9,400	9,700
GLT (MPa)	3,000	2,900	3,900
$v_{\mathit{LT}}$	0.28	0.32	0.30

Figure 5. Experimental versus CVSS-FEM failure modes of the validation case studies: (a) lateral-torsional buckling of I-LTB (experimental photo cited from Nguyen et al. (2014)), (b) web crippling of I-WC (experimental photo cited from Fernandes et al. (2015)), and (c) local buckling of I-LB (experimental photo cited from Yang et al. (2020)).

Table 4. Numerical versus experimental and theoretical buckling strength results of the validation case studies of I-shape FRP beams subjected to bending

Specimen ID	Test setup parameters	Theoretical buckling strength (MPa)^a (Bank, 2006)	Experimental buckling strength (MPa)	FEM buckling strength (MPa)
I-LTB	L/d = 33.33 a/d = N/A	${\sigma }_{cr}^{lat}=\frac{C_b}{S_x}\sqrt{\frac{{\pi }^2{E}_L{I}_y{G}_{\mathit{LT}}\mathit{J}}{{\left(k_f{L}_b\right)}^2}+\frac{{\pi }^4{E}_L^2{I}_y{C}_w}{{\left(k_f{L}_b\right)}^2{\left(k_w{L}_b\right)}^2}}=47.82$	46.87 Nguyen et al. (2014)	49.84
I-LB	L/d = 19.03 a/d = 6.56	${\sigma }_{cr}^{\mathrm{local}}=\frac{1}{{\left(\frac{b_f}{2}\right)}^2{t}_f}\left(7\sqrt{\frac{D_L{D}_T}{1+4.12{\zeta }_{\mathit{flange}}}}+12D_S\right)=173.98$	170.08 Yang et al. (2020)	181.85
I-WC	L/d = 2.00 a/d = N/A	${\sigma }_{cr}^{\mathrm{crippling}}=\frac{2{\pi }^2}{t_w{b}_{eff}^2}\left(\sqrt{D_L{D}_T}+D_{\mathit{LT}}+2D_S\right)=223.35$	221.24 Fernandes et al. (2015)	233.85

^a${\sigma }_{cr}^{lat}$ represents the critical lateral-torsional buckling stress. $C_b$ and $C_w$ are coefficients for moment variation and warping along the beam, respectively. $\mathit{J}$, $I_y$, and $S_x$ are the torsional constant, moment of inertia about the y-axis, and section modulus, respectively. $k_f$ and $k_w$ are the effective length factor for flexural and torsional buckling, respectively. $L_b$ is the unbraced length of the beam. $E_L$ and $G_{\mathit{LT}}$ are the longitudinal elastic modulus and in-plane shear modulus, respectively. ${\sigma }_{cr}^{\mathrm{local}}$ is the critical local buckling stress. $b_f$ and $t_f$ are the flange width and thickness, respectively. $D_L$, $D_T$, and $D_S$ are the longitudinal, transverse, and shear flexural stiffness components of the laminated profile. ${\zeta }_{\mathit{flange}}$ is the coefficient of restraint between the flange and web. ${\sigma }_{cr}^{\mathrm{crippling}}$ represents the critical web crippling stress. $t_w$ and $b_{\mathit{eff}}$ are the web thickness and depth, respectively.

Figure 6. Solution convergence of the CVSS analysis against default Abaqus/Standard analysis for the validation case studies.

Figure 7. Failure map showing the effect of span-to-depth ratio on I-shape pultruded FRP profiles subjected to three-point bending with different anisotropy ratios.

Figure 8. Failure map showing the effect of span-to-depth ratio on I-shape pultruded FRP profiles subjected to four-point bending with different anisotropy ratios ($\mathit{a}/\mathit{d}=\frac{1}{4}\mathit{L}/\mathit{d}$).

Figure 9. Failure map showing the effect of shear span-to-depth ratio on I-shape pultruded FRP profiles subjected to four-point bending with different anisotropy ratios ($\mathit{L}/\mathit{d}=40.0$).

Figure 10. Failure map showing the interaction between the span-to-depth ratio and shear span-to-depth ratio of I-shape pultruded FRP profiles subjected to four-point bending.

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Data availability statement

The data presented in this study are available on request from the corresponding author.