Behaviour of Reinforced Concrete Beams Under Bending Lecture (5) Behaviour of Reinforced Concrete Beams Under Bending DR: Ahmed Afifi Ass. Teacher: Ahmed Said

Stage of reinforced concrete beams failure Consider that a reinforced concrete beam as the one shown in figure 1 is subject to an increasing load that will cause the beam to fail. Several stages of behavior can be clearly identified. a: Before Cracking b: Cracking Stage, before yield, working load c: Ultimate and failure stage

Stage I: Cracking Stage 0 < M ≤ Mcr This stage begins from Zero loading till the first crack appears. At low load levels, below the cracking load, the whole of the concrete section is effective in resisting compression and tension stresses. In addition, since the steel reinforcement deforms the same amount as the concrete. It will contribute in carrying the tension stresses.at this stage, the distribution of strains and stresses are linear over the cross section shown in figure 2 Figure 2 : Before Cracking

Stage II: Working Stage Mcr < M ≤ Mw The working stage starts from the appearance of the first cracking till the crack width reaches 0.2 mm. when the load is further increased, the developed tensile stresses in the concrete exceed its tensile strength and tension cracks start to develop shown is figure 3. Most of these cracks are so small that they are not noticeable with the naked eyes. At the location of the cracks, the concrete does not transmit any tension forces ad steel bars are placed in the tension zone carry all developed tensile forces below the neutral axis. The neutral axis is imaginary line that separates the tension zone from the compression zone. Therefore, by definition the stress at the neutral axis is equal to zero’s the part of the concrete below the neutral axis is completely in the strength calculations and the reinforcing steel is solely responsible for resisting the entire tension force. At moderate loads (if the concrete stresses do not exceed approximately one third the concrete compressive strength), stresses and strains continue to be very close to linear. This called working loads stage shown in figure 3, which was the base of working stress design methods. Figure 3 : Working Stage , before yielding of steel

Stage II: Ultimate Stage Mw < M ≤ Mu This stage begins from crack width o.2 mm till steel reached yielding stresses. When the load is further increased, more cracks are developed and the neutral axis is shifted towards the compression zone. Consequently, the compression and tension forces will increase and the stresses over the compression zone will become nonlinear. However, the stress distribution over the cross section is linear. This called the ultimate stage shown in figure 4, the distribution of the stresses in the compression zone is the same shape of the concrete stress stain curve. The steel stress fs in this stage reaches yielding stress fy. For normally reinforced beams the yielding load is about 90-95 % of the ultimate load. At the ultimate stage two types of failure can be noticed. If the beam is reinforced with a small amount of steel, ductile failure will occur. In this type of failure, the steel yields and the concrete crushes after experiencing large deflections and lots of cracks. On the other hand, if the beam is reinforced with a large amount of steel brittle failure will occur. The failure in this case is sudden and occurs due to crushing of concrete in the compression zone without yielding of the steel and under relatively small deflections and cracks this is not a preferred mode of failure because it does not give enough warning before final collapse. Figure 4 : Ultimate Stage, ultimate load

Flexure Theory Assumption In order to analyze beam subject to pure bending certain assumptions have to be established. These assumptions can be summarized as follows: Strain distribution is assumed to be linear. Thus, the strain at any point is proportional to the distance from the neutral axis. This assumption can also be stated as, plane sections before bending remain plane after bending. The stain in the reinforcement is equal to stain in the concrete at the same level. The tension force developed the concrete is neglected. Thus, only the compression fore developed in the concrete is considered and all the tension force is carried with reinforcement. The stresses in the concrete and steel can be calculated using the idealized stress strain curves for the concrete and steel after applying the strength reduction factors. An equivalent rectangular stress block may be used to simplify the calculation of the concrete compression force. The above assumptions are sufficient to allow one to calculate the moment capacity of a beam. The first of these assumptions is the traditional assumption made in the development of the beam theory. It has been proven valid as long as the beam is not deep. The second assumption is necessary because the concrete and reinforcement must act together to carry the load and it implies a perfect bond between concrete and steel. The third assumption is obviously valid since the strength and the tensile force in the concrete below the neutral axis not affect the flexural capacity of the beam.

A. Stage I: Cracking Stage Calculations Calculating of the Cracking Moment Mcr :- n= 𝐸𝑠 𝐸𝑐 n=10 in this stage Ac = b x t Av = Ac + (n-1) As Neutral axis location (N.A): y’ = 𝑏 𝑥 𝑡 𝑥 𝑡 2 + 𝑛−1 𝐴𝑠 𝑥 𝑐𝑜𝑣𝑒𝑟 𝑏 𝑥 𝑡+ 𝑛−1 𝐴𝑠 Gross inertia of section about (N.A): Ig = 1/12 b t3 + b t (t/2 – y’)2 + (n-1) As (y’ – cover)2 Fctr = 0.6 x 𝐹𝑐𝑢 Mcr = 𝐹𝑐𝑡𝑟 𝑥 𝐼𝑔 𝑦′