Moment-curvature relationship of FRP-concrete-steel double-skin tubular members

Abstract

Tests were conducted on 3 specimens to study the flexural behavior of fiber reinforced polymer (FRP)-concrete-steel double-skin tubular members (DSTMs). The strip method was used to calculate the section moment-curvature curves of the 3 specimens and 12 models. A theoretical formula is presented for the flexural strength of DSTMs. The test results show that the tension zone of the specimen FRP tubes was in hoop compression while the compression zone was in hoop tension. The load-carrying capacity did not decrease even when the mid-span deflection reached about 1/24 of the span length. The tests, simulation and theoretical analysis resulted in a simplified formula for the flexural strength of DSTMs and a tri-linear moment-curvature model was expressed as a function of the section bending stiffness for DSTMs.

Keywords

fiber reinforced polymer (FRP) / concrete / steel / double-skin tubular members (DSTMs) / moment-curvature curve / flexural strength

Introduction

Fiber reinforced polymer (FRP)-concrete-steel double-skin tubular member (DSTM) is a kind of new structural member recently proposed by Teng et al. [1,2]. The member consists of an outer tube made of FRP tube [3] or wound by FRP sheets [1,2], an inner steel tube and the concrete between the two tubes. DSTM may be used as structural members that mainly withstand axial load. Elastic bending stiffness, post-yield bending stiffness and flexural strength are essential parameters for analyzing a structure, especially in the dynamic elastic-plastic analysis of the structure.

Based on the bending tests of 3 DSTMs and the numerical simulation and theoretical analysis of section moment-curvature curves of the 3 specimens and 12 models, the flexural behavior of DSTM was studied, a simplified formula of flexural strength for DSTM and a tri-linear moment-curvature model expressed as a function of section bending stiffness of DSTM were proposed.

Experimental work

The specimens were numbered as B1, B2, B3, and all had an overall length of 2 m. The outer tube of B1 was carbon fiber reinforcement polymer (CFRP) tube, the outer tubes of B2 and B3 were hybrid FRP tubes whose outside was 4 layers of glass fiber reinforcement polymer (GFRP) and inside was 6 layers of CFRP. The filament winding process was used to fabricate the GFRP tubes using S-glass fibers and the CFRP tubes using carbon fibers. The angle of the fibers of the FRP tubes measured with respect to the longitudinal axis of the tube was 80°. The fiber volume ratio of all the FRP tubes to epoxy resin was 75%. By applying a thin layer of epoxy to the inner surfaces of the FRP tubes, and spraying silicon sand on the top of the tacky epoxy, the interface between FRP tube and concrete core was roughened. Table 1 provides the parameters and mechanical properties of the FRP tubes and steel tubes. The diameter of the tubes was the inner diameter for the FRP tubes and the outer diameter for the steel tubes. D_o and t_FRP were the inner diameter and thickness of the FRP tubes. D_a and t_a were the outer diameter and thickness of the steel tubes. The mechanical properties of the FRP tubes, such as strength f, elastic modulus E, Poisson’s ratio ν, were obtained from the split-disk and axial compression tests of the rings which were cut from the hollow FRP tube, respectively. The values of strength f of the steel in Table 1 are the measured yield strength and the measured ultimate strength, respectively.

The center of the outer FRP tube of specimen B1 and B2 was consistent with the center of their inner steel tube. The inner steel tube of specimen B3 was eccentrically placed, i.e., its center was shifted towards the tension zone of the beam section. The parameters of the specimens are summarized in Table 2. In the table, the eccentricity e is the distance between the center of the steel tube and the center of the concrete section. The axial compressive strength f_c of concrete is taken as 0.76f_cu, where, f_cu is the measured concrete cubic strength. The fiber characteristic value is denoted by f_r/f_c, where, f_r is the maximum radial confining stress of FRP wrap, f_r=2f_ht_FRP/D_o, f_h is the equivalent hoop tension strength of FRP wrap. The ring specimens used to evaluate the hoop tension strength f_h were cut from the FRP tube. According to Ref. [4], f_h=1.75f_h,FRP, f_h,FRP is the hoop tension strength of FRP tube from split-disk test.

The test set up and the measurement locations are shown in Fig. 1. The specimens were placed horizontally and supported with a roller at both ends. The load was exerted on the distribute girder by a hydraulic jack. By the distribute girder, the middle part of the specimen was in pure bending. The load was measured by a load cell. The axial and hoop strains of the FRP tube were measured by strain gauges. The deflections at one third and mid-span of the specimen, the elongation at the middle of the specimen, and the slip among the FRP tube, steel tube and concrete core were measured by displacement transducers. The test data were collected by an IMP data acquisition system.

Test results and discussion

General observations

The tests were performed by a monotonic increasing load. When the load reached 45 kN, the sharp crack voice was heard; the obvious bending point was shown on the mid-span moment-deflection (M-δ) curve; and the bending stiffness of the specimen was decreased. The mid-span moments were calculated from the applied load. When the inner steel tube began to yield, no crack could be found on the FRP tube. With the load increasing, the first tensile crack was found near the loading point. When the load reached 70 kN, the compressive crack of the FRP tube was found. Until the end of the test, the mid-span deflection exceeded 1/24 of the span length; the load was not decreased. The specimen was very ductile.

The slip between the concrete and FRP tube was insignificant. The slip between the concrete and steel tube of specimen B1 was also insignificant, while that of specimen B2 and B3 was 2 mm and 5 mm, respectively. The slip between the concrete and steel tube was measured at the end of the specimen where the steel tube was not sealed by a steel plate. The reason is that the longitudinal shearing stress of the region, where subjected shear combined with bending, is larger than the bonding strength between the concrete and steel tube. With the increasing of space between the top of the steel tube and the top of concrete, the space of specimen B1, B2 and B3 was 25 mm, 38 mm and 56 mm, respectively, the shear stress at the steel tube-concrete interface increases, the slip between the concrete and steel tube is also increased. The failure mode of specimen B3 is shown in Fig. 2.

Mid-span section moment-deflection relationship

The mid-span section moment-deflection (M-δ) curves of the specimens are shown in Fig. 3. The outer FRP tube and inner steel tube of B2 and B3 were the same type respectively. Because the inner steel tube of specimen B3 was eccentrically placed, the tension area of the steel tube and the compression area of the concrete of specimen B3 were larger than those of specimen B2, the flexural strength of specimen B3 was higher than that of specimen B2.

Mid-span section moment-hoop strain relationship

The mid-span section moment-hoop strain (M-ϵ_h) curves of specimen B3 are shown in Fig. 4. In the figure, y is the distance between the strain gauge and the bottom of the beam, h is the height of the specimen, h=D_o+2t_FRP. Figure 4 indicates that, similar to the concrete filled steel tube beams [5], the compression zone of the FRP tube is in hoop tension and axial compression; the confinement of the FPR tube to the concrete is effective. As a result of Poisson’s effect, the tension zone of the FRP tube is in axial tension and hoop compression. However, the absolute value of the hoop tension strain at the top is much larger than that of the hoop compression strain at the bottom. Because of the compression strain gradient between the top of the beam and the neutral axis, the confinement of the FRP tube is reduced. A tension strain gradient between the bottom of the beam and the neutral axis also exists, but the FRP tube of the tension zone does not have confinement effect on the concrete.

Longitudinal strain distribution of mid-span section

The longitudinal strain distributions of the mid-span section at different load levels are shown in Fig. 5. In the figure, the vertical axis H is the distance between the strain gauge and the center of the FRP tube. The figures of graphic symbol are the ratio of applied load P to the maximum load P_max. Figure 5 indicates that, because there was no slip among steel tubes, the FRP tube and concrete occurred for specimen B1; on the whole, its strain distributions satisfy the plane section assumption. As a result of the slip between the steel tube and concrete, the tension strains at the bottom of the inner steel tube of specimen B2 and B3 (H=-57 mm in Fig. 5(b) and H=-75 mm in Fig. 5(c)), are much less than the tension strain computed by the plane section assumption. With the increment of slip, the strain distribution deviates more obviously from the plane section assumption.

Flexural strength

The bending capacities of the specimens are given in Table 3. In the table, M_m is the mid-span moment when the tests finished. The flexural strength is defined with four methods proposed in Refs. [5—7]:M_fs1 is the mid-span moment when the maximum compression strain of concrete reaches 0.0033; M_fs2 is the mid-span moment when the maximum tension strain of steel tube is 0.01; M_fs3 is the mid-span moment when the mid-span deflection is 1/50 of the span, i.e., 36 mm; M_fs4 is the mid-span moment when the deflection caused by the pure bending zone at the mid-span section, meaning the deflection at mid-span section minus the deflection at loading point, is 1/100 of pure bending span, i.e., 6 mm.

Section moment-curvature curves

The section moment-curvature curves of DSTMs reflect their flexural behavior. From the curves, the variation of bending stiffness of DSTMs could be studied.

Calculation of section moment-curvature curves

Based on the plane section assumption, the strip method was used to write a program to calculate the section moment-curvature curves of the specimens. In the program, the second model given by Ref. [3], i.e., the curve ② in Fig. 6, was used for concrete stress-strain relationship. The formulas used to calculate the model parameters can be obtained from Ref. [3]. The tension force of the concrete is quite small compared with that of the steel tube, for simplicity and with little if any error, the contribution of the concrete and FRP tube in tensile zone is ignored. The compressive zone of the FRP tube is in hoop tension and axial compression; the orthogonal anisotropic elastic stress-strain model proposed in Ref. [8] was adopted. From the Tsai-Hill failure criterion [8], if the axial compression stress of the FRP tube in the compressive zone is larger than the axial compression strength of the FRP tube, it will be out of work. In the program, the values of axial compression elastic modulus, hoop tension elastic modulus and Poisson’s ratio were obtained from the split-disk tests and the axial compression tests.

The test results and numerical analysis results of section moment-curvature curve of 3 specimens are shown in Fig. 7. In the figure, the values of curvature were obtained by the measured displacements at the top and bottom of the specimens. The flexural strengths M_fs (section moment corresponding to the maximum compression strain of concrete of 0.0033) are given in Table 4. The results indicate that for the specimens with no slip between concrete and steel tube, such as specimen B1, the calculated flexural strength and moment-curvature curves agree well with the test results. If the slip between concrete and steel tube occurs, such as specimen B2 and B3, the calculated curve is higher than the test curve because of the overestimation of the tensile stress of the steel tube according to the plane section assumption. As shown in the figure, when the moment of specimen B2 reaches 21 kN•m, the applied load drops a little. After that, however, the specimen still sustains an increased load with the increasing of deformation. In practice, many measures could be taken, such as to seal two ends of the inner steel tube by steel plate, to weld hoop steel anchor bars or shear studs on the inner steel tube, and to evidently reduce the slip between concrete and steel tube. The effect of slip on the section moment-curvature relationship of DSTMs need to be further studied.

Nominal flexural strength

Similar to concrete filled steel tube beams [6], the section moment-curvature curves of DSTMs can be divided into 3 branches, i.e., elastic branch, elastic-plastic branch and hardening branch. Referring to Refs. [6,7] that concrete evidently is confined by FRP tube was taken as the beginning of hardening stage, the flexural strength was defined as the moment when the strain of the outermost fiber of concrete reached 0.0033. The following basic assumptions were used to derive the section nominal flexural strength of DSTMs [7,9,10]: 1) the plane section of specimens remains plane after bending; 2) ignoring the tension strength of concrete and FRP tube; 3) the strain of the outermost fiber of concrete is 0.0033; 4) the compressive stress diagram of concrete is an equivalent rectangle. From the stress-strain relationship model given in Ref. [3], if ϵ_u reaches 0.0033, the FRP tube has little effect on the confinement of concrete. By analyzing the test results, the equivalent compression strength f'_cm of concrete is (1+0.2f_r/f_c)/f_c; 5) the ratio β of depth of rectangular stress block x to the depth of neutral axis x_n varies with the variation of relative depth of compression zone ξ_n. If ξ_n≤0.5, then β=0.8; if 0.5<ξ_n≤0.75, then β=1.0667-0.2667ξ_n [10].

From the above assumptions, the stress and strain distribution diagrams at flexural strength are given in Fig.8. In the figure, θ is the central angle, the subscripts x, n, ex, ax', ax denote the equivalent compression depth of the concrete relative to the center of the FRP tube, the compression depth of the core concrete relative to the center of the FRP tube, the compression depth of the steel tube relative to the center of the steel tube, the compressive plastic depth of the steel tube relative to the center of the steel tube, and the tensile plastic depth of the steel tube relative to the center of the steel tube, respectively; ξ is the equivalent relative depth of compression zone; h₀=r_o+r_a+e is the section effective depth; ϵ'_y and ϵ_y are the steel tube compression and tension yield strain, respectively; f'_a and f_a are the steel tube compression and tension yield strength, respectively; r_o and r_a are the inner radius of the FRP tube and the outer radius of the steel tube, respectively ; γ'=f'_a/(0.0033E_a) and γ=f_a/(0.0033E_a) are the ratios of steel compression and tension yield strain to the strain of outermost fiber of concrete ϵ_u, respectively; σ_c,FRP is the axial compression stress of the FRP tube.

Taking f'_a=f_a from the equilibrium of section axial force and moment, the following formulas can be obtained:

(1)N=C_{\mathrm{1}}f{\prime }_{cm}{A}_c+C_{\mathrm{2}}{f}_a{A}_a+C_{\mathrm{3}}{\sigma }_{c,FRP}{A}_{FRP},

(2)M_{fs}{=C}_4{f}_{cm}^{}{A}_c\xi h_0{+C}_5{f}_a{A}_a(r_a+e{)+C}_6{\sigma }_{c,FRP}{A}_{FRP}(r_o+e),

where A is the area of cross section, subscripts c, a, FRP denote concrete, steel tube and FRP tube, respectively;

\begin{array}{c}{C}_1=\left[\right(2{\theta }_{\mathrm{x}}-\sin 2{\theta }_{\mathrm{x}})r_{\mathrm{o}}^2-(2{\theta }_{\mathrm{ex}}-\sin 2{\theta }_{\mathrm{ex}}\left)r_{\mathrm{a}}^2\right]/\left[2\pi \right(r_{\mathrm{o}}^2-r_{\mathrm{a}}^2\left)\right],\\ C_3=(\sin {\theta }_{\mathrm{n}}-{\theta }_{\mathrm{n}}\cos {\theta }_{\mathrm{n}})/\left[\pi \right(1-\cos {\theta }_{\mathrm{n}}\left)\right],\\ C_4=\left[0.5\right(r_{\mathrm{o}}^3-r_{\mathrm{a}}^3)-2r_{\mathrm{o}}\cos {\theta }_{\mathrm{n}}(r_{\mathrm{o}}^2-r_{\mathrm{a}}^2)+2r_{\mathrm{o}}\cos {\theta }_{\mathrm{n}}(r_{\mathrm{o}}^2\cos {\theta }_{\mathrm{x}}-r_{\mathrm{a}}^2\cos {\theta }_{\mathrm{ex}})+2r_{\mathrm{a}}^{2}e(1-\cos {\theta }_{\mathrm{ex}})\\ -0.5(r_{\mathrm{o}}^3\cos 2{\theta }_{\mathrm{x}}-r_{\mathrm{a}}^3\cos 2{\theta }_{\mathrm{ex}})]/[\pi \xi h_0(r_{\mathrm{o}}^2-r_{\mathrm{a}}^2)],\\ C_6=({\theta }_{\mathrm{n}}-1.5\sin 2{\theta }_{\mathrm{n}}+2{\theta }_{\mathrm{n}}{\cos }^2{\theta }_{\mathrm{n}})r_{\mathrm{o}}/\left[2\pi \right(1-\cos {\theta }_{\mathrm{n}}\left)\right(r_{\mathrm{o}}+e\left)\right].\end{array}

If r_a-e≤γ'ξ_nh₀, then

$$\begin{gathered} C_{\mathrm{2}}=[\mathrm{0.0033}{E}_{\mathrm{a}}{\gamma }^{}(\sin {\theta }_{\mathrm{ax}}-{\theta }_{\mathrm{ax}}\cos {\theta }_{\mathrm{ax}})+f_{\mathrm{a}}(\sin {\theta }_{\mathrm{ax}}+\pi \cos {\theta }_{\mathrm{ax}}-\pi -{\theta }_{\mathrm{ax}}\cos {\theta }_{\mathrm{ax}})]/[f_{\mathrm{a}}\pi (\mathrm{1}-\cos {\theta }_{\mathrm{ax}})], \\ {C}_{\mathrm{5}}=r_{\mathrm{a}}[(\mathrm{0.0033}{E}_{\mathrm{a}}{\gamma }^{}+f_{\mathrm{a}})·({\theta }_{\mathrm{ax}}-\mathrm{0.5}\sin \mathrm{2}{\theta }_{\mathrm{ax}}-\mathrm{2}\sin {\theta }_{\mathrm{ax}}\cos {\theta }_{\mathrm{ex}}+\mathrm{2}{\theta }_{\mathrm{ax}}\cos {\theta }_{\mathrm{ax}}\cos {\theta }_{\mathrm{ex}}) \\ +\mathrm{2}\pi f_{\mathrm{a}}\cos {\theta }_{\mathrm{ex}}(\mathrm{1}-\cos {\theta }_{\mathrm{ax}})]/[\mathrm{2}\pi (\mathrm{1}-\cos {\theta }_{\mathrm{ax}})(r_{\mathrm{a}}+e)]. \end{gathered}$$

If r_a-e>γ'ξ_nh₀, then

$$\begin{gathered} C_{\mathrm{2}}=({\theta }_{\mathrm{ax}}+{\theta }_{\mathrm{a}{\mathrm{x}}^{}}-\pi )/\pi , \\ {C}_{\mathrm{5}}=r_{\mathrm{a}}[{\theta }_{ax}-{\theta }_{{\mathrm{ax}}^{\text{'}}}+\mathrm{2}\cos {\theta }_{\mathrm{ex}}(\sin {\theta }_{{\mathrm{ax}}^{\prime }}-\sin {\theta }_{\mathrm{ax}}+{\theta }_{\mathrm{ax}}\cos {\theta }_{\mathrm{ax}}-{\theta }_{{\mathrm{ax}}^{\text{'}}}\cos {\theta }_{{\mathrm{ax}}^{\text{'}}})]/[\pi (\cos {\theta }_{\mathrm{a}{\mathrm{x}}^{}}-\cos {\theta }_{\mathrm{ax}})(r_{\mathrm{a}}+e)] \\ +(\sin {\theta }_{{\mathrm{ax}}^{\text{'}}}+\sin {\theta }_{\mathrm{ax}})r_{\mathrm{a}}/[\pi (r_{\mathrm{a}}+e)]+r_{\mathrm{a}}\cos {\theta }_{\mathrm{ex}}/[\pi (r_{\mathrm{a}}+e)]. \end{gathered}$$

If DSTMs are subjected to pure bending, taking N equals to 0, the equivalent relative depth of the compression zone ξ can be derived, C4, C5, C6 and flexural strength can then be given.

The equations of C4, C5 and C6 are comparatively complex. Simplified formulas of flexural strength of DSTMs were proposed. As has been stated, if there is no slip between steel tube and concrete, the computed results of strip methods will agree well with the test results. Then, the section moment-curvature curves of 3 specimens and 12 models were computed by the strip method. The parameters of 3 specimens and 12 models, and the computed results of flexural strength are given in Table 4. Regressing the computed results, the following simplified formulas are given: ξ=f_aA_a/(f'_cmA_ck⁴_v+3.3f_aA_a+0.5f_c,FRPA_FRP), C₄=ξ(r_a+e)/r_o, C₅=1/(1+1.3ξ), C₆=0.088f_c,FRP/σ_c,FRP; f_c,FRP is the axial compression strength of the FRP tube.

Then, a simplified formula of flexural strength M_fs of DSTMs is given by

(3)M_{fs}=f{\prime }_{cm}{A}_c{\xi }^{\mathrm{2}}{h}_{\mathrm{0}}(r_a+e)/r_o+f_a{A}_a(r_a+e)/(\mathrm{1}+\mathrm{1.3}\xi )+\mathrm{0.088}{f}_{c,FRP}{A}_{FRP}(r_o+e)\mathrm{.}

The average ratio of the results calculated by Eq. (3) to the results computed by the strip method is 1.013 with a standard deviation of 0.04. The two methods are in good agreement.

Section moment-curvature curve model for double-skin tubular members

A tri-linear moment-curvature (M-ф) model is adopted. The slopes of three branches of the model are section elastic bending stiffness, elastic-plastic stiffness and hardening stiffness. To regress the test results and the strip method analysis results, equations of the tri-linear moment-curvature model can be expressed as

(4)

where, EI is the section bending stiffness of DSTMs, I is the section moment of inertia, M is the section moment. The moment-curvature curves of the specimens computed by Eq. (4) are shown in Fig. 7.

Conclusions and recommendations

1) The bending test results of the three DSTM specimens show that the tension zone of the FRP tube is in hoop compression and axial tension, tension cracks appear, the compression zone of the FRP tube is in hoop tension and axial compression, and epoxy resin shear cracks appear. The mid-span deflection of specimens exceeds 1/24 of the span, while the strength does not decrease.

2) Placing the inner steel tube of DSTM eccentrically, i.e., to shift it towards the tension zone of the section, can improve the flexural strength of DSTMs.

3) The section moment-curvature curves computed by the strip method with the plane section assumption and confined concrete stress-strain relationship, agree well with the test results when no or little slip between the steel tube and concrete occurs, but is higher than the test results when the slip is comparatively large.

4) The proposed formulas of flexural strength and tri-linear moment-curvature model expressed with section bending stiffness of DSTMs agree well with the strip method results, and can apply to DSTMs with no slip between the steel tube and concrete. The effect of slip on the section moment-curvature relationship of DSTMs needs to be further studied.

In practice, to avoid the occurrence of slip among steel tube, FRP tube and concrete, two ends of the inner steel tube should be sealed with steel plates; hoop steel anchor bars or shear studs should be welded on the inner steel tube. The inner surface of FRP tubes, i.e., the interface between the FRP tube and concrete core, should be roughened by applying a thin layer of epoxy, and spraying silicon sand on the top of the tacky epoxy.

References

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