Plastic mechanism analis o CHS stub columns strengthened using CFRP M. Elchalakani School o rchitectural, Civil and Mechanical Engineering, Victoria University, Melbourne, ustralia M.R. Bambach Department o Civil Engineering, Monash University, Melbourne, ustralia BSTRCT: This paper presents a plastic mechanism analis or circular hollow section (CHS) tubes strengthened using carbon iber reinorced polymer (CFRP) deorming in an axi-symmetric (elephant oot) collapse mode under large deormation axial loading. The collapse proceeded progressively by olding about three concentrated hinge lines and hoop extension o the shell. n expression or the plastic collapse axial load was obtained by equating the total energy absorbed in bending and extension to the external work carried out during deormation o the tube. The newly derived mathematical model takes into account the contribution o the CFRP towards energy absorption during collapse. Comparisons o the predicted instantaneous post-buckling collapse loads with those obtained rom experiments carried out elsewhere show good agreement. 1 INTRODUCTION Hollow steel tubes are widely used as columns in many structural stems and a common ailure mode o such tubes when subjected to axial compression (or combined loading) is local buckling near a column end. For example hollow steel tubes are used as bridge biers in Japan and such piers suered extensive damage and even collapse during the 1995 Hyogoken-nanbu earthquake [1]. One o the methods used to in seismic retroit o such bridge piers is the use o CFRP rapping (see Fig 1). Thus, it is important to predict the response o the composite section (CHS+CFRP) under large deormation axial compression. Plastic mechanism analis is widely used to derive such predictions. (b) nti-symmetric mode o collapse (c) Elephant oot with a racture (a) Typical Column Figure 1. Damaged bridge piers in Japan [1]. 1
The collapse mechanisms o tubular stub columns were studied in the past by Johnson et al [], bramowicz and Jones [3], and Meng et al [4]. Wierzbicki bramowicz [5] used velocity iled approach to derive general ormulations or the crushing o thin-walled structures. The basic olding o an isolated plate orming the roo mechanism was studied by Davies et al [6] and Mahendran [7]. Ohkubo et al [8] provided an expression or the mean crushing load o hat sections commonly used in the utomotive industry where they showed that the radius o the rolling hinge has signiicant eect on such load. Mamalis et al [9] studied non-metallic plastic square tubes under axial load. The eect o CFRP on the collapse o composite circular tubes was studied by Mamalis et al [1], Song et al [11], Gupta and bbas [1], Hanei and Wierzbicki [13] and more recently by Wang and Lu [14]. However, these precious crush studies derive ormulations or the mean crush load and little research ocused on the development o the collapse curves such as Key and Hancock [15], Zhao et al [16], Grzebieta [17] and more recently Elchalakani et al [18, 19] or bending o CHS. Grzebieta [17] developed ormulation or the instantaneous axial collapse load versus axial delection or an empty CHS orming an axi-symmetric collapse mode. It is shown in this paper that his rigid plastic mechanism is modiied to include the eects o the inite length o the plastic hinges and the CFRP strengthening. The newly derived collapse curves and those developed by Grzebieta s [17] will be compared against experimental collapse curves obtained recently by Teng and Hu [1]. Neutral axis t t s t x σ σ y o o σ Figure. Stress distribution in composite tube wall. σ ( t x) = σ ( x t ) + σ t (1) σ ( t + t ) σ t s x = () σ To evaluate the ull plastic moment per unit width to bend plastically the composite section M p we take moment about the neutral axis ( t x) ( x t ) t M p = σ + σ + σ y t ( x ) (3) Introducing the dimensionless ratios k =σ y / σ and t r = t / t s, Equation 3 can be written as σ ts M p = (1 + ktr + ktr k tr ) (4) 4 Expressing the ull plastic moment M p in a relation to equivalent yield stress or the ace ( σ ) ' 1+ k tr + k tr k tr σ y.σ (1 + tr ) y = (5) ' y PREDICTION OF COLLPSE CURVES.1 Eect o CFRP Consider the bimetallic plate (shown in Fig. ) consisting o ully adhering dierent materials o thickness ts (or CHS lats) and t or (FRP sheet) and corresponding yield stresses σ and σ y, respectively. Examining the ull yielding o the composite cross section o overall thickness t = t s + t and unit width and requiring the orces on the composite section we obtain the position o the plastic neutral axis x rom. Work dissipated by bending The key mechanical properties o the mild steel specimens reinorced using normal modulus CRP sheets tested in [1] are listed in Table 1. The geometry o the CHS and the external CFRP sheets are shown in Figure 3. The elastic modulus o the steel and CFRP sheet was 1 and 8.1 GPa, respectively. The stub columns were about 5 mm long and tested under uniorm axial compression. The theoretical expressions or the energy components consumed in bending and hoop stretching are developed below and combined to give the overall load-axial displacement relationship or the composite section. 5 Table 1. Details o specimens [1]. Spec. D t s σ σ y t t r mm mm MPa MPa mm (=t /t s ) ST-F 165 4. 333... ST-F1 166 4. 333 185.17.4 ST-F 165 4. 333 185.34.81 ST-F3 166 4. 333 185.51.11
t CFRP t P R D t s Figure 3. Geometry o SHS+CFRP. Figures 4 and 5 show the outward olding mechanism and its kinematics, respectively, developed at large axial deormations. The deormation energy consumed during bending at the three plastic hinges in Fig 5 (, B and B) o the outward olding mechanism can be expressed as dw = 4 π ( R + R M d (6) b B ) p in which is the angular change deined in Figure 5, R and R B are the radii o the deormed tube at the plastic hinges and B, and M p = σ y t /4. Note that R =R and ρ = L / and L(1 cos) R B = R + ρ (1 cos) = R + (7) Using Equation 7, Equation 6 can be simpliied to L(1 cos) dwb = 4 πm p R + d (8) where is the angular change at the plastic hinge or B in Figure 5. L is the hal-wave length o the elephant oot mechanism which was assumed constant during the course o deormation..3 Work dissipated by hoop extension s the tube is compressed, the tube expands in the hoop direction. The deormation energy or the elephant oot due to this expansion can be given by dw h = πσ θ tl cosζdζ (9) Knowing that ζ = / and assuming ull composite action in the hoop direction using Equation 5, ie, σ θ = σ y, where σ θ is the hoop stresses, γ is an angle deined in Figure 5 and ρ is the current radius. Thus Equation 9 can be simpliied rom the kinematical relations given in Figure 5 to ζ B L/ C L- δ max P Figure 4. The theoretical model. dw 4 πσ y tl sin ( / ) cos( / ) = d (1) h Most o the work carried out by the internal orces have now been determined and can be summed and equated to the work carried out by those orces external to the structure (dw ext = dw int ). This external energy can be expressed as dw ext = P. dδ (11) sin δ = L (1 ) (1) sin cos dw ext = P. dδ = PL d (13) where δ is the axial displacement and is the total angular change shown in Figure 5. Thereore, denoting the increment o rotation d, the instantaneous post buckling collapse load can be written as L(1 cos 4πM p R + P = L(sin cos) C E ' C' B' 4πσ y tl sin ( / ) cos( / ) L(sin cos) + B D F (14) 53
L B ρ (1-cos ) ζ ρ γ γ B R = R ρ R B= R+ ρ (1-cos ) L(1 cos tσ y R + + πttl P = 3 L(sin cos) 4σ L sin ( / ) cos( / ) (15) Note L = 1. 347 Rt was adopted in Grzebieta [17] and subsequently was used in Equation 15. slightly larger average value o L = 1.843 Rt was used in Equation 14. larger hal-wave length was used in the numerical modelling perormed by Teng and Hu [1] to match the experimental collapse curves or the plain and strengthened CHS specimens. dδ/ L +d ρ (1-cos( +d )) R = R ρ ζ+d B ζ γ -dγ +d γ -dγ B' R B = R+ ρ (1-cos(+d)) Figure 5. Mechanism kinematics. Note, the contribution o the CFRP is depicted in the 1 st and nd terms in Equation 14 which represents the energy consumed in the bending and hoop extension, respectively. Note, in the tests the CFRP sheets were rapped in the hoop direction, where it has no strength in the axial direction. The collapse curves can be plotted by irst assuming a value or and substituting in Equation 14, the value o P can be readily determined. From the geometrical compatibility given in Equation 1, δ can be computed. similar Equation can be derived by slightly modiying Equation 3 in Reerence [17] by assuming ull composite action in the hoop direction as ollows 1 Test ST-F Test ST-F (Teng and Hu 7) New Model-Plain Grzebieta (199)-Plain 5 1 15 Figure 6. nalis results or ST-F 3 RESULTS OF THE NLYSIS The collapse curves predicted using both Equations 14 (the New Theory) and 15 (Grzebieta [17]) are plotted in Figures 6 to 9 together with the experimental curves or all the specimens tested recently by Teng and Hu [1]. Test specimen ST-F represents a plain CHS with no strengthening. Specimens ST- F1, ST-F and ST-F3 represent CHS strengthened with one, two and three layers o CFRP sheets, respectively. It is seen that there is a good agreement in the slope o the collapse curve and values o the post-buckling axial load particularly at large axial deormations. It is also seen that Grzebieta [17] model or plain CHS signiicantly underestimates the axial load, but its modiied version (Equation 15) provides reasonable predictions. It is seen that Equation 15 under estimates the axial load compared to Equation 14. It appears that adding more layers ater Reer ppendix or Figure 7-9 54
strengthening with layers o CFRP does not have signiicant eect on strength. This is also shown correct by the new model. 4 CONCLUSIONS This paper presented a plastic mechanism analis or circular hollow section (CHS) tubes strengthened using carbon ibre reinorced polymer (CFRP) deorming in an axi-symmetric (elephant oot) collapse mode under large deormation axial loading. The collapse proceeded progressively by olding about three concentrated hinge lines and hoop extension o the shell. n expression or the plastic collapse axial load was obtained by equating the total energy absorbed in bending and extension to the external work carried out during deormation o the tube. The newly derived mathematical model takes into account the contribution o the CFRP towards energy absorption during collapse. Comparisons o the predicted instantaneous post-buckling with those obtained rom experiments carried out elsewhere show good agreement. REFERENCES 1. Teng, J.G. and Y.M. Hu, Behaviour o FRP- Jacketed Circular Steel Tubes and Cylindrical Shells under xial Compression. Construction and Building Materials, 7. 1(9): p. 87-838.. Johnson, W., Soden, P. D. and S.T.S. l- Hassani, Inextensional Collapse o Thin- Walled Tubes under xial Compression. Journal o Strain nalis, 1977. 1(4): p. 317-33. 3. bramowicz, W. and N. Jones, Dynamic Progressive Buckling o Circular and Square Tubes. International Journal o Impact Engineering, 1984a. 4(4): p. 43-7. 4. Meng, Q., l-hassani, S. T. S. and P.D. Soden, xial Crushing o Square Tubes. International Journal o Mechanical Science, 1983. 5(9-1): p. 747-773. 5. Wierzbicki, T. and W. bramowicz, On the Crushing Mechanics o Thin-Walled Structures. Journal o pplied Mechanics, 1983c. 5: p. 77-734. 6. Davies, P., Kemp, K. O. and.c. Walker, n nalis o the Failure Mechanism o an xially Loaded Simply Supported Steel Plate. 55 Proceedings o Institute o Civil Engineers, 1975. 5(Part ): p. 645-658. 7. Mahendran, M., Local plastic mechanisms in thin steel plates under in-plane compression. Thin-Walled Structures, 1997. 7(3): p. 45-61. 8. Ohkubo, Y., kamatus, T. and K. Shirasawa, Mean Crushing Strength o Closed-Hat Section Members. Society o utomotive Engineers, 1974. Paper 74: p. 1-1. 9. Mamalis,.G., Manolakas, D.E., Baldoukas,.K. and G.L. Viegelahn, The xial Crushing o Thin Walled Steel PVC Tubes and Frusta o square Cross Section. International Journal o impact Engineering, 1989b. 8(3): p. 41-64. 1. Mamalis,.G., Manolakas, D.E., Demosthenous, G.. and Johnson, W., xial Plastic Collapse o Thin Bi-Material Tubes as Energy Dissipating Stems. International Journal o impact Engineering, 1991. 11(): p. 185-196. 11. Song, H.-W., et al., xial Impact Behaviour and Energy bsorption Eiciency o Composite Wrapped Metal Tubes. International Journal o impact Engineering,. 4(): p. 385-41. 1. Gupta, N.K. and H. bbas, Lateral Collapse o Composite Cylindrical Tubes between Flat Platens. International Journal o impact Engineering,. 4(): p. 39-346. 13. Hanei, E.H. and T. Wierzbicki, xial Resistance and Energy bsorption o Externally Reinorced Metal Tubes. Composites, Part B, 1996. 7B: p. 387-394. 14. Wang, X. and G. Lu, xial Crushing Force o Externally Fibre-Reinorced Metal Tubes. Proceedings o the Institute o Mechanical Engineers,. 16(863-873). 15. Key, P.W. and G.J. Hancock. Plastic Collapse Mechanisms or Thin-Walled Cold Formed Square Tube Columns. in 1th ustralian Conerence on the Mechanics o Structures and Materials. 1986. The University o delaide. 16. Zhao, X.-L., B. Han, and R.H. Grzebieta, Plastic mechanism analis o concrete-illed double-skin (SHS inner and SHS outer) stub
columns. Thin-Walled Structures,. 4(1): p. 815-833. 17. Grzebieta, H., n alternative Method or Determining the Behaviour o Round Stocky Tubes subjected to an xial Crush Load. Thin-Walled Structures, 199. 9: p. 61-89. 18. Elchalakani, M., X.L. Zhao, and R.H. Grzebieta, Plastic mechanism analis o circular tubes under pure bending. International Journal o Mechanical Sciences,. 44(6): p. 1117-1143. 19. Elchalakani, M., Zhao, X. L., Grzebieta, H., Plastic Collapse nalis o Slender Circular tubes Subjected to Large Deormation Pure Bending. dvances in Structural Engineering -n International Journal a. 5(4): p. 41-57. 56
ppendix 1 Test ST-F1 (Teng and Hu 7) New model-composite Grzebieta (199)-Composite ppendix Test ST-F1 5 1 15 Figure 7 nalis results or ST-F1 1 Test ST-F (Teng and Hu 7) New Model-Composite Grzebieta (199) - Composite Test ST-F 5 1 15 Figure 8 nalis results or ST-F 1 Test ST-F3 (Teng and Hu 7) New Model-Composite Grzebieta (199) - Composite Test ST-F3 5 1 15 Figure 9 nalis results or ST-F3 57