simply supported composite beams is concerned these phenomena occur at all cross-sections, and so have little influence on distributions of bending moment. Differently,in composite beams significant tension in concrete occurs only in hogging regions. It is influenced by the sequence of construction of the slab and by effects of temperature. shrinkage,and longitudinal slip. The fle al riai of a fully cracked composite section can be as low as quarter of the uncracked value e variation in flex eds to uncertaintyt isibution of longitudinal along a tinuous beam of uniform section.This moments,and hence in the amount of cracking to be expected. 3.Design principle and calculation method There are two ways to determine internal moments in continuous composite beams:elastic analysis rigid-plastic analysis. The rigid plastic analysis can be applied depending on the susceptibility of the beam to local bucklina.so i must check the class of the steel section in order to know if it allow for plastic ution or not can always be applied,but it is quite inefficient because it produces higher moments at inte al suppor than inmid-panbu e resistance of a continuous beam in negative bending at internal supports is usually less than the resistance in the mid d-span.Thus a moment redistribution is used to evaluate the correct load carrying capacity of continuous composite beams. Then moment redistribution is used,and it allow for: the inelastic behavior that occurs in all materials in a composite beam before maximum load is reached: the effects of cracking of concrete at serviceability limit states ifying the bending-moment distribution found for a particular loading while um beween the actionsand the bending moments Monentar at cross-sections where the ratio of action effect to resistance is highest.The effect is to increase the moments of opposite sign.Design Codes commonly permit negative(hogging)moments at supports to be reduced,except at cantilevers,by redistribution to mid-span. 4.Moment redistribution according to Elastic analysis EuroCode 4 [2]suggests some redistribution values for both"cracked"and"uncracked"analysis. From EuroCodes 4(UNI-EN-1994-1-1_2005_EEN): The bending moment distribution given by a linear elastic global analysis according to 5.4.2 may be redistributed in a way that satisfies equilibrium and takes account of the effects of inelastic behaviour of materials,and all types of buckling.(.)
simply supported composite beams is concerned these phenomena occur at all cross-sections, and so have little influence on distributions of bending moment. Differently, in composite beams significant tension in concrete occurs only in hogging regions. It is influenced by the sequence of construction of the slab and by effects of temperature, shrinkage, and longitudinal slip. The flexural rigidity of a fully cracked composite section can be as low as quarter of the ‘uncracked’ value, so a wide variation in flexural rigidity can occur along a continuous beam of uniform section. This leads to uncertainty in the distribution of longitudinal moments, and hence in the amount of cracking to be expected. 3. Design principle and calculation method There are two ways to determine internal moments in continuous composite beams: elastic analysis rigid-plastic analysis. The rigid plastic analysis can be applied depending on the susceptibility of the beam to local buckling, so I must check the class of the steel section in order to know if it allow for plastic redistribution or not. The elastic analysis can always be applied, but it is quite inefficient because it produces higher moments at internal support than in mid-span regions, but the resistance of a continuous beam in negative bending at internal supports is usually less than the resistance in the mid-span. Thus a moment redistribution is used to evaluate the correct load carrying capacity of continuous composite beams. Then moment redistribution is used, and it allow for: - the inelastic behavior that occurs in all materials in a composite beam before maximum load is reached; - the effects of cracking of concrete at serviceability limit states. It consists of modifying the bending-moment distribution found for a particular loading while maintaining equilibrium between the actions and the bending moments. Moments are reduced at cross-sections where the ratio of action effect to resistance is highest. The effect is to increase the moments of opposite sign. Design Codes commonly permit negative (hogging) moments at supports to be reduced, except at cantilevers, by redistribution to mid-span. 4. Moment redistribution according to Elastic analysis EuroCode 4 [2] suggests some redistribution values for both “cracked” and “uncracked” analysis. From EuroCodes 4 (UNI-EN-1994-1-1_2005_EEN): “The bending moment distribution given by a linear elastic global analysis according to 5.4.2 may be redistributed in a way that satisfies equilibrium and takes account of the effects of inelastic behaviour of materials, and all types of buckling.(…)
Table 5.1:Limits to redistribution of hogging moments,per cent of the initial value of the bending moment to be reduced Class of cross-section in hogging moment region 1 2 3 4 For un-cracked analysis 40 30 20 10 For cracked analysis 25 15 0 Fig.3Table 5.1 from Eurocode4 The bending moments in composite beams determined by linear elastic global analysis may be modified.Different kind of redistribuition exists,depending on the class of the cross section,on the state(cracked or uncracked),and the prescription changes according to different codes. Nealectina crackina of concrete Eurocode 4 121 allow for a redistribution from 40%to 10%of ant from the ho nding of sections gging rec the redistribution varies from 5 to%always depending on the clasific ge of m of the cross section A similar approach is also proposed in the Chinese Design Code [1],but it proposes a redistribution only for 15%of the moment for the uncracked section analysis.This prescription is more conservative with respect to Eurocodes. Then the allowed moment redistribution in the two design codes is quite different,given that a redistribution of 40%is allowed in Eurocode 4 [2],which is much larger than 15%allowed by Chinese code [1]. 5.Moment redistribution according to Rigid Plastic Analysis Rigid plastic analysis allow for a more efficient design approach for continuous composite beams A key factor for this design method is the rotation capacity,which must be sufficient to allow a moment redistribution.The rotation capacity can be defined as the sum of the effects of non- linear behaviour of the component materials in positive and negative moment regions of the beam The actual rotation capacity should be greater than the required rotation capacity at the rde the bution the dep of the stee ction that is in compression and the lateral moment region.The low ratio of negative-to-positive moment resistance in composite beams allow for a greater redistribution of moment,and then for a greater rotation capacity,required to develop a plastic collapse mechanism.In continuous composite beam it is considerably larger than in a continuous steel beam where the moment ration is greater. In order to study the required moment redistribution based on rigid plastic analysis,a two span continuous composite beam of uniform section under uniform distributed loading is considered
Fig. 3 Table 5.1 from Eurocode 4 The bending moments in composite beams determined by linear elastic global analysis may be modified. Different kind of redistribuition exists, depending on the class of the cross section, on the state (cracked or uncracked), and the prescription changes according to different codes. Neglecting cracking of concrete Eurocode 4 [2] allow for a redistribution from 40% to 10% of moment from the hogging regions to the sagging regions depending on the classification of the cross sections in hogging. While for cracked analysis the percentage of moment redistribution varies from 25% to 0% always depending on the classification of the cross section. A similar approach is also proposed in the Chinese Design Code [1], but it proposes a redistribution only for 15% of the moment for the uncracked section analysis. This prescription is more conservative with respect to Eurocodes. Then the allowed moment redistribution in the two design codes is quite different, given that a redistribution of 40% is allowed in Eurocode 4 [2], which is much larger than 15% allowed by Chinese code [1]. 5. Moment redistribution according to Rigid Plastic Analysis Rigid plastic analysis allow for a more efficient design approach for continuous composite beams. A key factor for this design method is the rotation capacity, which must be sufficient to allow a moment redistribution. The rotation capacity can be defined as the sum of the effects of nonlinear behaviour of the component materials in positive and negative moment regions of the beam. The actual rotation capacity should be greater than the required rotation capacity at the notional hinge in order to ensure the process of moment redistribution. The proportion of the depth of the steel section that is in compression and the lateral slenderness ratio strongly influences the available rotation capacity in the negative moment region. The low ratio of negative-to-positive moment resistance in composite beams allow for a greater redistribution of moment, and then for a greater rotation capacity, required to develop a plastic collapse mechanism. In continuous composite beam it is considerably larger than in a continuous steel beam where the moment ration is greater. In order to study the required moment redistribution based on rigid plastic analysis, a two span continuous composite beam of uniform section under uniform distributed loading is considered
工 L BA L C elastic analysis MM ultimate state Mp n LLnL☐ Fig.4Two span continuous composite beam Based on the plastic mechanism theory,the ultimate distributed load can be expressed as: 2M w= (1) Where:M is the plastic moment resistance in thes M'is the plastic agging the ita me m the side support n=(W1+-1)/ u=M/Mp is the ratio of the negative on positive moment resistance The hogging moment at the internal support can be computed,in term of a linear static uncracked analysis,as follow: Ma -gwpL2 (2) From this formula it is possible to evaluate the distributed load: M'.8 w=+B亚 (3) (1)and (3): %a微→=1- M'a8 (4) And so B is the redistribution required to develop a full plastic mechanism. Fig.3shows the variation of the moment redistribution required with respect to the ratio ofM to Mp.The moment redistribution required B,increases asudecreases. For an equal spanned continuous composite beam of uniform cross section,u is normally less than 1.0,say 0.7 for example,then 47%moment redistribution at the internal support is required to enable a plastic design
Fig. 4 Two span continuous composite beam Based on the plastic mechanism theory, the ultimate distributed load can be expressed as: Where: is the plastic moment resistance in the sagging moment region is the plastic moment resistance in the hogging moment region is the distance from the side support is the ratio of the negative on positive moment resistance The hogging moment at the internal support can be computed, in term of a linear static uncracked analysis, as follow: From this formula it is possible to evaluate the distributed load: Where is the moment redistribution ratio, which can be derived from equating the equations (1) and (3): And so is the redistribution required to develop a full plastic mechanism. Fig. 3 shows the variation of the moment redistribution required with respect to the ratio of to . The moment redistribution required increases as decreases. For an equal spanned continuous composite beam of uniform cross section, μ is normally less than 1.0, say 0.7 for example, then 47% moment redistribution at the internal support is required to enable a plastic design
0.7ff M.R.-moment redistibution 0.6 8 required M.R. 40%M.R..EC4 0.3 0.2 25%M.R.in Chinese Design Code 0.1 0 0.5 06 0.7 0.8 0.9 1 H=M 'IMp Fig.5 Variation of B with u Now I'm going to consider a continuous beam with different span length and different uniformed load in each mes. span.The hogging moment based on elastic analysis becom M'a= (5) The load carrying capacity of the beam each span is expressed as: 8M(1+入。m) M=1-)1+2wan (6a) 8Mp(1+Aspan)AwAspan w收=aPa+九wa月 (6b) At ULS,the notional plastic hinge is favorable in the longer and the heavier load span which will govern the moment redistribution required in developing a plastic mechanism.So the required moment redistribution of each span is derived and expressed as: A4=1-4-A-11+pa （1+wpan)μ (7a) A2=1-4-0+久 (7b) (1+w23nan)μ
Fig. 5 Variation of with Now I’m going to consider a continuous beam with different span length and different uniformed load in each span. The hogging moment based on elastic analysis becomes: Where: is the ratio of the two different uniformed loading is the ratio of the different span length The load carrying capacity of the beam each span is expressed as: At ULS, the notional plastic hinge is favorable in the longer and the heavier load span which will govern the moment redistribution required in developing a plastic mechanism. So the required moment redistribution of each span is derived and expressed as:
r B. m-1.0 、m-0.75 02 0.1 04 5 0.6 07 0.8 0.9 1 H-MIM, Fig.6 Requiredn tredistribution for different spans,whenw,w 0.7r8. 03 &=05 01 0.50.60.70.80.9 1 H=MM。 Fig.7 Required moment redistribution for different lod intwo span, The previous graphs show the variation of the moment redistribution for different ratio of spans and different ratios of load in the two spans.it is possible to see that the required moment distribution increases as u decreases,as seen for the p loads.On the other hand the e Brevious beam,with equal spans and eduat the difference nm increas difference of lo tbeam with equa spans and eqa unifo s increases distributed load,a greater moment redistribution is required. Determination of moment redistribution at the ultimate limit state To make sure that a plastic mecha nism shall occur,a cor omposite beam should posses hinge r the internal support. Two different situ capa city at the pote tions may occurs in this kind of beams,even if the beams has plastic cross section along the full span If the available moment redistribution is less than the required moment redistribution(B<B) the ultimate limit failure of the beam is governed by the hogging moment resistance and the rotation ductility at the internal support.This is due to the fact that the rotation capacity at the potential hinge in the hogging region is insufficient to develop a full plastic mechanism for the beam,and so o the failure oc urs i this region,while the sagging moment in the span fails to reach the plastic :moment resistance. The load carrying capacity in this situation can be computed by:
Fig. 6 Required moment redistribution for different spans, when Fig. 7 Required moment redistribution for different load in two span, The previous graphs show the variation of the moment redistribution for different ratio of spans and different ratios of load in the two spans. It is possible to see that the required moment distribution increases as decreases, as seen for the previous beam, with equal spans and equal loads. On the other hand the required moment redistribution decrease as the span difference increases, or as the difference of load in the two spans increases. It can be deduced that in a continuous composite beam with equal spans and equal uniformed distributed load, a greater moment redistribution is required. Determination of moment redistribution at the ultimate limit state To make sure that a plastic mechanism shall occur, a continuous composite beam should posses sufficient rotation capacity at the potential hinge near the internal support. Two different situations may occurs in this kind of beams, even if the beams has plastic cross section along the full span. If the available moment redistribution is less than the required moment redistribution ( ) the ultimate limit failure of the beam is governed by the hogging moment resistance and the rotation ductility at the internal support. This is due to the fact that the rotation capacity at the potential hinge in the hogging region is insufficient to develop a full plastic mechanism for the beam, and so the failure occurs in this region, while the sagging moment in the span fails to reach the plastic moment resistance. The load carrying capacity in this situation can be computed by: