Design and Analysis of Beams

1. Design and Analysis of Beams 1) Analysis of Reinforced Concrete Sections for Flexure • A beam is a structural member that supports applied loads and its own weight primarily by internal moments and shears. Fig-2.1 shows a simply supported beam that supports its own dead weight and an applied load P. If the axial applied load, N, is equal to zero as shown, the member is referred to as a beam. If N is a compressive force, the member is called a beam-column. If it were tensile, the member would be a tension tie. This chapter is restricted to the very common case where N = 0.

2. Analysis of Reinforced Concrete Sections for Flexure

3. Analysis of Reinforced Concrete Sections for Flexure

4. Analysis and Design of Singly Reinforced Rectangular Beams General points to be considered in design of RC beams: • Due to materials non - homogeneity and whether there is a normal force or not, the neutral axis in RC cross- section is not in general expected to coincide with the geometric centroid. • Due to its inherent behavior of cracking, concrete is weak in resisting tension and normally the tensile capacity of concrete is neglected and thus the over all tension over the cross-section is taken care of by the reinforcement steel. • The general assumption “plane section remain plane” is assumed to be complied with implying that strain distribution over the x-section is linear and the strain in the reinforcement is equal to the strain in the concrete at the same location.

5. Analysis and Design of Singly Reinforced Rectangular Beams • Concrete cracks due to tension as a result; reinforcement is required where flexure, axial loads, or shrinkage effects cause tensile stresses. Therefore it is important that designers be able to project and visualize the different shapes of the structure and identify tension zones. The reinforcement bars for flexure are placed on the tensile face of the member that is on the tension side of the deflected shape. • The width of the beam is mostly decided or chosen from the functional service or recommended practices. • Depth of the beam should be chosen in such a way that undesirable and excessive deflections are avoided. Cods provide provision for minimum effective depths of beams.

6. Analysis and Design of Singly Reinforced Rectangular Beams • Necessary concrete cover and bar spacing should be considered due to the following reasons • To bond the reinforcement to the concrete so that the two elements act together. • To protect the reinforcement against corrosion. • To protect the reinforcement from strength loss due to overheating in the case of fire. • The arrangement of bars within a beam must allow sufficient concrete on all sides of each bar to transfer forces into or out of the bars. Fresh concrete can be placed and compacted properly around all the bars if sufficient space is provided.

7. Analysis and Design of Singly Reinforced Rectangular Beams • Usually RC beams crack at lower bending moments and even though the moment is small, once the cracking moment is exceeded, a sudden failure could occur with little or no warning. For this reason minimum flexural reinforcement should be provided. The amount of minimum flexural reinforcement is usually specified in building codes. • Once the geometric dimension, h, b, d and the material quality fck, fy are chosen and decided, the amount of reinforcement can be determined by considering equilibrium condition of the moment capacity of the assumed section and the design bending moment. In doing so it would be a good practice to assume that the reinforcement steel has yielded and reached a stress level of fyd.

8. Analysis and Design of Singly Reinforced Rectangular Beams Stress-strain diagrams Figure 2.2 shows the idealized (characteristic and design) stress-strain relationships for concrete in compression. The basic shapes of the curves, i.e. parabolic-rectangular. The ultimate strain (εcu2) of the concrete in compression is taken as 0.0035 and εc2= 0.002. Fig-2.2 Parabolic-rectangular stress-strain diagram for concrete

9. Analysis and Design of Singly Reinforced Rectangular Beams The design steel stresses, fyd, are derived from the idealized (characteristic) stresses, fyk, by dividing by the partial safety factor for steel, γs: Figure 2.3 shows the idealized and design stress strain curves for reinforcing steel recommended in EBCS 2. Fig. 2.3 Design stress strain diagram for steel reinforcement

10. Analysis and Design of Singly Reinforced Rectangular Beams Consider a rectangular beam section as shown below: Fig-2.4 Section with strain diagram and stress blocks

11. Analysis and Design of Singly Reinforced Rectangular Beams Modes of Failures of Beams Due to Flexure Depending on the properties of a beam, flexural failures may occur in three different ways: Tension failure: Reinforcement yields before concrete crushes (reaches its limiting compressive strain) Such a beam is said to be under reinforced Compression failure: Concrete crushes before steel yields Such a beam is said to be over reinforced Balanced Failure: Concrete crushes and steel yield simultaneously Such a beam has balanced reinforcement