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Lecture_8 - (2 Moment redistribution of reinforced concrete...

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Unformatted text preview: (2) Moment redistribution of reinforced concrete beam sections • Once a RC beam section develops its ultimate moment of resistance Mu , it then behaves as a plastic hinge (Fig. 2.3-6) due to (1) the plastic behaviour of concrete (Fig. 2.1-1), and (2) the elasto-plastic behaviour of steel (Fig. 2.1-2). Fig. 2.3-6 Plastic hinge in a reinforced concrete section o In an indeterminate RC structure, once the plastic hinge forms at a section, further loading must be taken by other parts of the structure, with changes in moment elsewhere. o The plastic behaviour of plastic hinges is limited by the concrete failure; or more specifically, the concrete failure limits the rotation that may take place at a section in bending. This is because concrete will fail at a relatively small compressive strain. o Available rotation of a hinge does not cause crushing of the concrete, and further hinges may be formed until a mechanism is produced. o A full plastic analysis, such as has been developed for structural steel, could then be applied to reinforced concrete beam design. 62 • Three basic requirements for applying moment redistribution: (1) Equilibrium between internal and external forces must be maintained. (2) At sections with the largest moments, the depth of neutral axis should satisfy (Clause 3.2.2.1) x ≤ ( β b − 0.4) d (2.32) This provision ensures that there is adequate rotation capacity at the section for redistribution to take place. βb is not to be taken as exceeding 0.9, i.e. β b ≤ 0.9 , when moment redistribution of the section is considered. (3) The moment of resistance of any section should be at least 70 percent of the moment from the elastic analysis; hence the code (Clause 3.2.2.1) allows a reduction of up to 30% of the peak elastic moment to be made, i.e. the maximum moment redistribution permitted is 30 percent. Thus, β b ≥ 0.7 . • From the requirements (2) and (3): 0.7 ≤ β b ≤ 0.9 . o For 0-10% moment redistribution: βb = 0.9 For 20% moment redistribution: βb = 0.8 For 30% moment redistribution: βb = 0.7 o From Eq. (2.32), the corresponding depth of neutral axis x = 0.3 to 0.5 d 63 • The requirement (3) (allowing up to 30% moment redistribution) is o To prevent an excessive demand on the ductility and rotational capacity. A 15% moment redistribution is generally to be taken as a reasonable limit, though BS 8110 permits up to 30%. (The moment redistribution ≤ 10% in general design practice.) o To ensure that there can be no movement in the position of points of contraflexure obtained from the elastic analysis (Fig. 2.3-7), ensuring that a sufficient length of tension steel is provided to resist cracking at the serviceability limit state. Fig. 2.3-7 Redistribution of hogging moments Fig. 2.3-8 shows an internal span in a continuous beam where the peak elastic moment at the support is reduced by 30%, and the sagging moment is increased accordingly. (a) Fig. 2.3-8 30% redistribution for support moments in an internal span in a continuous beam. (a) Loading; (b) moment diagrams 64 (b) Fig. 2.3-8 (cont’d) 30% redistribution for support moments in an internal span in a continuous beam. (a) Load; (b) moment diagrams The moments at service loads are about 1/1.5 = 0.7 of the elastic moments at ultimate loads, where 1.5 is the average safety factor for dead and imposed loads. The elastic moment diagram for service loads shows a length of beam in tension, which would be in compression under the redistribution moments. There is generally sufficient reinforcement in the top of the beam to resist the small moment in this area. • Example 2.3-4 Moment redistribution for a single-span fixed-end beam The beam shown in the figure is subjected to an increasing uniformly distributed load. It is assumed that the ultimate bending strengths (Mu) at the mid-span and the supports are the same and equal to wL2/12, where w is the UDL to cause the first plastic hinge. Determine (1) the collapse load, W, the beam may carry with moment redistribution; (2) the moment redistribution ratio at section C. 65 i) Elastic support moments = wL2/12 Elastic span moment = wL2/24 ii) At collapse with consideration of moment redistribution MA = MC = Mu = wL2/12 MB = Mu = wL2/24 + waL2/8 where waL2/8 is the additional moment at B, as for a simply supported beam with hinges at A and C. Thus, wL2/12 = wL2/24 + waL2/8 Hence wa = w / 3 ≈ 0.33w iii) Collapse load the beam may carry with moment redistribution W = w + wa = 1.33w iv) When the beam is subjected to W, the moment redistribution ratio at C is βb = wL2 wL2 = = 0.75 WL2 1.33wL2 66 • In the design practice, ο The elastic moment diagram can be obtained for the required ultimate loading in the ordinary way. ο Some of these moments (usually at supports of a continuous beam) may be reduced; but this will necessitate increasing others to maintain the static equilibrium. ο The moment redistribution leads to a distribution of bending moments away from peak moment regions, resulting in reducing congestion of steel reinforcement at such regions. giving a possible saving in amount of reinforcing steel required. 67 Example 2.3-5 Moment redistribution for a continuous beam (Example 7.4 in Reference book by MacGinley and Choo) (a) Elastic analysis for three cases of load combinations Fig. 2.3-9 Loads, bending moments and shear force diagrams. (a) Case 1; (b) Case 2; (c) Case 3. 68 Fig. 2.3-10 (a) Shear force envelope; (b) bending moment envelope. (b) Moment redistribution ο Consider Case 1 from Fig. 2.3-9. The hogging moments at supports B and C are reduced by 20% (to 215.88 kNm). The redistributed moment diagram is drawn in Fig. 2.3-11. * Note the length in tension due to hogging moment from the redistribution. Moments - kNm Fig. 2.3-11 Bending moments in case 1 before and after moment redistribution 69 ο Using the reduced value for moments at interior supports B and C, the redistribution of moments is also carried out for Case 2 and Case 3. The shear force and bending moment diagrams for all three cases can be drawn and shown in Fig. 2.3-12. The shear force and bending moment envelopes are then constructed, as shown in Fig. 2.3-13. Fig. 2.3-12 Shear force and bending moment diagrams after moment redistribution: (a) Case 2; (b) Case 3. 70 Fig. 2.3-13 Internal force envelopes after moment redistribution. (a) Shear force envelope; (b) bending moment envelope. ο Comparing Figs 2.3-10 and -13 shows that both the maximum hogging and sagging moments from the elastic bending moment envelope have been reduced by the moment redistribution. The redistribution gives a saving in the amount of steel reinforcement required. ο Fig. 2.3-13 shows the redistributed moment does not meet the code requirement that the ultimate resistance moment must not less than 70% of elastic maximum moment near the internal supports. The modified redistributed moment diagram is shown in Fig. 2.3-13(b). 71 ...
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This note was uploaded on 12/24/2011 for the course CIVL 232 taught by Professor Jskuang during the Spring '06 term at HKUST.

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