INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2011 Copyright 2010 All rights reserved Integrated Publishing services Research article ISSN 0976 4399 A parametric study of RC moment resisting frames at joint level by investigating Moment -Curvature Former Student, Department of Civil Engineering, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh. sifatmuin@gmail.com doi:10.6088/ijcser.00202010090 ABSTRACT Nonlinear analysis can evaluate actual performance of a structure during an earthquake. Among various simplified nonlinear analysis procedures and approximate methods to estimate maximum inelastic displacement demand of structures pushover analysis is a significant one. To perform pushover analysis moment curvature (M-φ) relation at hinge level needs to be defined. M-φ relation of RC frames are complex in nature due to a unique bonding of two materials with completely different mechanical properties. In this paper, numerically derived M-φ relation of beams and columns is used to investigate the effect of different parameters on it. This paper will be helpful in deciding which parameter to alter and the amount of it to get expected strength and ductility. It will also be useful to designers in defining M-φ relation during pushover analysis. Keywords: Ductility, Moment curvature relation, crushing curvature, confined concrete, moment capacity, transverse reinforcement spacing. 1. Introduction Earthquake is a global phenomenon. It causes significant damage every year in different part of the world. It has been a field of interest for the researchers to minimize the loss of life and property due to such catastrophe. Static Nonlinear performance based analysis is the result of such researches. It is a vast tool that is applied to simulate structural performance during an earthquake. In recent times, Guner & Vecchio (2010), D Ambrisi et al. (2009), Pereira et al. (2009), Guo et al. (2011) all have used nonlinear pushover analysis in their research. In this method, status of damage is indicated by hinges formed in the frame elements. Therefore, nonlinear moment-curvature behavior are assigned to discrete locations along the length of frame (line) elements. However, moment curvature relation is dependent on various parameters and hence it is essential to know the effect of these parameters on M-φ curves before applying them in pushover analysis. Research have been conducted to find out these effect on steel (Ricart & Plumier, 2008).This paper aims to study the joint M-φ of RC frames and investigate the effect of different parameters on them. 2. Description of Model Received on June 2011 published on September 2011 23
INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 1, 2011 Copyright 2010 All rights reserved Integrated Publishing services Research article ISSN 0976 4399 To analytically create a moment-curvature hip for any section, assumptions must be made for material stress-strain hips. Several equations have been suggested to Received on June 2011 published on September 2011 24
model this nonlinear stress-strain hip (f- ) of concrete. Among them, the Kent & Park (1971) model which was later extended by Scott et al. (1982) has been chosen in this work because of its applicability for confined concrete also. It approximates the stress-strain hip by a parabola up to the ultimate strength (f c ) and a straight line beyond that up to the crushing of concrete (Figure1). Figure 1: Stress-Strain Relationship of Confined and Unconfined Concrete. On the other hand, concrete has been assumed to behave linearly in tension up to the concrete modulus of rupture. A simple elastic-perfectly plastic model will be assumed for the reinforcing steel in tension or compression, i.e., it is assumed to be linear up to the yield point, beyond which the stress is assumed to be constant. 3. Moment-Curvature Relationship of RC sections & Verification The M-φ hip of RC sections is derived numerically by the application of simple principles of strength of materials. By following the procedure described by Olivia & Mandal (2005) a computer program has been developed. In order to check the accuracy and limitations of the M-φ curves generated by the program CSI-Column is used. A 30cm x 60cm beam and a 30cm x 30cm column with axial force of 450kN and 1000kN is analyzed with both CSI-Column and numerical program. For the beam, CSI-COL generated M-φ curve completely overlaps the numerically generated curve (Figure 2). However, crushing curvature φ u is lower in the CSI-COL curve. This is due to the fact that, crushing strain ε u is assumed to be 0.003 in CSI-COL according to ACI code. Whereas, ε u is calculated to be 0.00367 by the proposed concrete model for this section and it varies with crushing strength of concrete. Similarly, for columns with axial force, curves plotted by CSI-COL almost overlaps the curves produced by the program.(figure 3 ). In these case also, the difference in crushing curvature is visible which is caused by the difference in crushing strain. Similar, verification has been done by comparing the curves with curves presented by Anam & Soma (2002) and Noor (2010). 24
Figure 2: Verification of the M-φ curve of Beam Figure 3: Verification of the M-φ curve of Colum with P=1000kN 25
4 Parametric Studies A section behaves differently when subjected to only flexure (beam) and combined action of flexure and axial force (column). Therefore a beam section and a column section have been modeled to study the effects of different parameters. Material and sectional properties of the sections are summarized in Table 1. Table 1: Sectional properties of beam and column Section Property Beam Column Width b 30cm 30cm Height h 45cm 30cm Concrete strength fc 28 MPa 28 MPa Steel Grade f y 275 MPa 275 MPa Reinforcement ratio ρ 0.01 0.02 Cover 5 cm 4 cm Axial Force P 0 450kN 4.1 Effect of Transverse Reinforcement Spacing As a result of confinement effect of transverse reinforcement ultimate strain increases and therefore higher ultimate curvature is found. Here spacing is considered according to provisions for SMRF (d/4 for beam & 10cm for column), OMRF (s=20cm) with two other cases. Figure 4: M-φ curves of beam for different spacing 26
In case of beam, enough ductility is available even without hoops or with widely spaced hoops (Fig. 4).In case of columns, sections without stirrups have low ductility. Therefore, stirrup spacing plays an important role in improving ductility of the section (Fig. 5). Figure 5: M-φ curves of column for different spacing 4.2 Effect of Reinforcement Ratio Figure 6: M-φ curves of beam for different ρ 27
Figure 7: M-φ curves of column for different ρ In order to observe the change in M-φ relation due to change in reinforcement ratio ρ in beams three values of ρ (=0.01, 0.015, 0.02) have been taken. Figure 6 shows M- φ curves for each ρ. From the figures it can be seen that as ρ increases moment capacity increases but ductility decreases. This is due to decrease in ultimate curvature as more concrete is active in resisting moment. For column higher values of ρ has been selected (ρ= 0.02, 0.04, 0.06). With increase in ρ moment capacity increases in column as well. But ductility remains almost unchanged. Same amount of concrete is active in resisting axial force in all the cases. 4.3 Effect of Steel Grade Effect of steel grade on M-φ curve of beam is shown in Figure 8. Moment capacity increases and ductility decreases with increase of steel grade. For higher steel grade tension force is higher at ultimate strain and more concrete is active in moment. Ductility therefore decreases. Column behaves similarly due to change in f y of steel (Fig. 9). The change in both moment and ductility factor is quite significant. 28
Figure 8: M-φ curves of beam for different f y 4.4 Effect of Concrete Strength Figure 9: M-φ curves of Column for different f y Figure 10 shows the effect of concrete crushing strength f c on M-φ relation. It is evident that with the increase of f c moment capacity increases slightly but ultimate curvature 29
decreases. Similarly, for columns Figure 11 shows that with the increase of f c strength increases to some extent but no definite trend is observed in ductility. Table 5 shows this discrepancy in ductility factor in both beam and column. This results from the fact that ultimate strain of concrete ε u depends on f c Therefore when f c increases ε u and with that φ u decreases. But for a given strain φ increases with increase of f c These opposite actions causes the complications in determining effect of f c on M-φ. Figure 10: M-φ curves of beam for different f c Figure 11: M-φ curves of column for different f c 4.5 Effect of Axial Force on Column M-φ 30
M-φ curve of a column section depends on the axial force it is subjected to. When failure is governed by yielding, fully developed M-φ curves are formed. But when section fails due to dominance of axial force, M-φ curves are not well defined (Fig. 12). Here, four different values of axial force have been selected from four zones of interaction curve. As the axial force increases maximum moment increases but ductility reduces significantly. 5. Conclusion Figure 12: M-φ curves of column for different P. In this paper effect of different parameters on M-φ curve have been studied. It is observed that amount of steel, steel grade and spacing of transverse reinforcement have significant effect on M-φ of a section. When amount of longitudinal steel and steel grade increases moment increases but ductility decreases. Whereas, closely spaced transverse reinforcement improves ductility without changing moment capacity. Columns are found to be less ductile in nature than beams due to effect of axial force. In seismic design one objective is to ensure that sections have enough strength to withstand the massive force induced by earthquake. Therefore high amount of reinforcement are provided. From the analysis, it is evident that it reduces the ductility of the section. However, providing adequate ductility is another objective in seismic design. In this situation, spacing of transverse reinforcement becomes a governing criteria in design of earthquake resistant structures. Therefore, high strength materials along with closely spaced transverse reinforcement is recommended to ensure adequate strength and ductility of a section situated in an earthquake prone zone. 6. References 1. ACI Committee 318 (2002), Building Code Requirement for Structural Concrete, American Concrete Institute, Detroit. 31
2. Anam, I. and Shoma, Z.N., (2002), Nonlinear Properties of Reinforced Concrete Structures. 2nd Canadian Conf. on Nonlinear Solid Mech., Vancouver, Canada, 2, pp 657-666. 3. D Ambrisi A., Stefano M. D. and Tanganelli M., (2009), Use of Pushover Analysis for predicting seismic response of irregular buildings: A case study. Journal of Earthquake Engineering, 13(8), pp 1089-1100. 4. Guner S. and Vecchio F.J. (2010), Pushover Analysis of shear- critical frames: verification and application. ACI Structural Journal, 107(1), pp 72-81. 5. Guo L., Uang C., Elgamal A., Prowell I. and Zhang S. (2011), Pushover Analysis of a 53m high wind turbine tower. Advanced Science Letters, 4, pp 656-662. 6. Kent, D.C. and Park, R. (1971), Flexural members with confined concrete. ASCE Journal of Structural Division, 97(7), pp 1969-1990. 7. Noor, M. A. (2010), Designing with Grade 500 Steel. The University Press Ltd. Dhaka, Bangladesh. 8. Olivia, M. and Mandal, P. (2005), Curvature Ductility of Reinforced Concrete Beam. Journal of Civil Engineering, 6(1), Indonesia. 9. Pereira V.G., Barros R.C. and Cesar M.T. (2009), A Parametric study of a RC frame based on Pushover Analysis. 3 rd International Conference on Integrity, Reliability and Failure, Portugal 10. Ricart L.S. and Plumier A. (2008), Parametric study of ductile momentresisting steel frames: A first step towards Eurocode 8 calibration. Earthquake Engineering and Structural Dynamics, 37(7), pp 1135-1155. 11. Scott, B.D., Park,R. and Priestley, M.J.N. (1982), Stress-Strain Behavior of Concrete Confined by Overlapping Hoops at Low and High Strain Rates. ACI Journal, 79(1), pp 13-27. 32