Introduction of beam
Beam is the structural member which is primarily subjected to bending moment or shear stress. It supports load and their own self weight by internal moment and shear. In limit state, beam is firstly designed for strength and durability & their performance is then checked with regards to other limit of serviceability., example deflection and cracking.
The failure of beam occurs when either steel or concrete reaches limiting strain. The limiting strain of steel is higher than that of concrete so actual collapse of the actual member is always caused by crushing of concrete.
- Balanced section
In balanced section, the maximum stress in concrete and steel reaches to their permissible stresses at the same time. So the balanced failure is expected to occur by the simultaneous initiation of concrete and yielding of stress at the same time.
2. Under-reinforced section (tension or ductile failure)
In under-reinforced section, less steel is used than that required for the balanced section. So, the steel is stressed to its maximum permissible stress while concrete is below the permissible stresses. Hence the primary causes of failure is yielding in tension steel. The onset of failure is gradual, giving ample proper prior warning of impending collapse by the way of increased curvature, deflection and cracking. The large increase in curvature prior to collapse is the indication of typical ductile mode of failure.
3. Over-reinforced section (compression or brittle failure)
In over-reinforced section, more steel is used than that required for balanced section. So, the steel is not fully stressed to its maximum permissible tensile stress while compression stresses in concrete at the extreme fiber reaches its maximum permissible value. The failure of concrete is sudden without warning and deflection and curvature remains low right up to failure.
Note: Hence for the full utilization of material and economic design, the balanced section is taken as the best section. For seismic point of view, the under-reinforced section has ductile failure and gives enough proper prior warning with large deflection curvature. So under-reinforced section is more preferable. But for the design, it is better to avoid the over-reinforced section as the failure is brittle.
Before going to singly and doubly reinforced beam, let’s see the assumptions of limit state of collapse for flexure:[IS 456:2000 clause 38.1]
- The plane section normal to the axis remains plane after bending.
- The tensile strength of concrete is neglected.
- The maximum strain in concrete at the outermost fiber is taken as 0.0035 in bending.
- The maximum strain in the tensile reinforcement in the section at the failure shall not be less than fy/ (1.15 Es) +0.002
- The stress-strain curve for the concrete in compression section may be assumed to be rectangle, trapezoid, parabola or any other shape which result in prediction of strength in substantial agreement with the test results.
- There exist a perfect bond or adhesion between concrete and reinforcing steel up to the point of failure so no slippage between the two materials.
The concrete member should show an acceptable stress-strain curve for design purposes. The stress distribution for strain varying from 0 – 0.002 is parabolic and for strain 0.002 – 0.0035 is constant (so has a rectangular shape).
The compressive strength of concrete in structure shall be assumed to be 0.67 times the characteristics strength. And a partial safety factor 1.5 shall be applied. Thus the maximum stress in design curve which corresponds to design compressive strength of concrete is 0.447 fck. The partial safety factor for steel ϒm = 1.15 shall be applied.
Singly reinforced beam:
The beam which is designed only for tensile reinforcement is called singly reinforced beam. In this types of beam, only tensile reinforcement is provided so the compression is resisted by the concrete only. It is easy to design due to its simple calculation. Despite no steel bar is designed for compression side, minimum 2 number of bar should be provide to support traverse reinforcement.
Doubly reinforced beam:
Reinforced concrete beam in which steel reinforcement is designed for both tension and compression side is called doubly reinforced beam. If required area of steel is more than the limiting area of steel for balanced section (ie. steel required for balanced section in singly reinforced), compression steel is provided to increase the moment carrying capacity.
Compression steel increases the amount of curvature before failure due to increase in ductility. The code IS 13920:1993, clause 6.2.3 recommend minimum of 50% of tension steel to be provided as compression steel.
Compression steel is also effective in reducing long-term deflection due to shrinkage and creep. We can clearly visualize this by the use of ANNEX C of IS 456:2000 for calculating defection and shrinkage.
Compression reinforcement must be provided if the bending moment at a section changes its sign due to dynamic loads (occurs in the span of continuous beam in bridge girder or in beam subjected to lateral load).
The design of doubly reinforced beam is always economical to design and reduces the self-weight as it decrease the dimension.
How to know what to use? Either singly or doubly reinforced?
In case of size restriction and for that dimension, if it can’t resist the required flexural moment then we shall have to go for doubly reinforced beam.
Design for shear:
The beam may be subjected to shear in addition to the predominant flexure. Despite bending moment and shear force exists together, they are designed separately according to traditional design philosophy. The actual design for shear is to be based on principal stresses developed. A practical design procedure is presented in code IS 456:2000 based on the average or nominal shear stresses across the section.
Nominal shear stress:
- The nominal shear stress in beam of uniform depth shall be obtained by following equation:
Ⴀv = Vu/bd
2. The nominal shear stress in beam of varying depth shall be obtained by following equation:
Ⴀv = (Vu ±Mu/d * tanᵦ) / bd
Ⴀv = nominal shear stress
Vu = shear force due to designed load
b = breadth of the member
d = effective depth
Mu = bending moment at the section
ᵦ = angle between the top and bottom edges of the beam
The negative sign in the formula applied when the bending moment Mu increases numerically in the same direction as the effective depth d increases, and vice versa.
- When nominal shear stress Ⴀv is less than the design shear strength of concrete Ⴀc in table 19 of code IS 456:2000 then minimum shear reinforcement shall be provided in accordance with clause 22.214.171.124 of code IS 456:2000 .
- When the nominal shear stress Ⴀv exceeds the design shear strength of concrete Ⴀv in table 19, then shear reinforcement shall be provided in any of the following forms:
- Vertical stirrups
- Bent-up bars along with stirrups
- And inclined stirrups
However code specifies, where bent-up bars are provided, their contribution towards shear resistance shall not be more than half that of the total shear reinforcement.
When more than one type of shear reinforcement is used to reinforced the portion of the beam, the total shear resistance shall be computed as the sum of the resistance for various types separately.
The area of the stirrups shall not be less than the minimum as the specified clause 126.96.36.199 of code IS 456:2000 and specify to avoid the compression failure of the section in shear.
Beam subjected to torsion:
When the structure is subjected to torsion, twisting of the plane occurs and such collapse in torsion also occurs when we are designing in limit state of design such that such failure occurrence is called the limit state of collapse in torsion.
A plain cutting in the concrete beam of the rectangular cross-section subjected to torsion develops spiral cracks and fails suddenly. In RC member subjected to flexure and torsion, the flexural cracks are first formed and just after that the flexural and torsional stiffness is formed on the beam face.
The beam subjected to torsion combined with flexure and shear, does not require determination of torsional reinforcement separately from that required for flexure and shear.