Balanced sections,under reinforced section and over reinforced section
Balanced Sections
A balanced sections is that in which stress in concrete and steel reach their permissible value at the same time. This means that stress diagram is as shown in Fig. 2.6(b). The percentage of steel corresponding to this section is called as balanced steel and the neutral axis is called as critical neutral axis nc
\[\frac {m.\sigma_{cbc}}{\sigma_{st}}=\frac {n_{c}}{d-n_{c}}\]
For a balanced sections, the moment of resistance is calculated as under :
\[ M _{B}=\frac{\sigma _{cbc}}{2}b.n _{c}\left ( d-\frac{n _{c}}{3} \right )=Rbd^{2}\]
Under Reinforced Section
In an under reinforced section, the percentage of steel provided is less than that provided in balanced section. So the actual neutral axis will shift upwards i.e., nc > n as shown in Fig. 2.6(c). In under reinforced section, the stress in steel first reaches it permissible value, while the concrete is under stressed. The moment of resistance of this section is calculated as
\[M _{r}=\sigma _{st}.A _{st}\left ( d-\frac{n}{3} \right )\]
The various features of under reinforced section are as follows :
(i) Steel is fully stressed while concrete not (i.e., stress in steel is σst (permissible) but stress in concrete is less than σcbc
(ii) The actual neutral axis lies above the critical neutral axis (n < nc).
(iii) The percentage of steel is less than the balanced section hence the section is economical.
(iv) Ductile failure.
(v) The moment of resistance is less than balanced section.
In under reinforced section, the failure is ductile because steel fails first and sufficient warning is given before collapse. Due to ductile failure and economy, the under-reinforced sections are preferred by designers.
Over Reinforced Section
In an over reinforced section the percentage of steel provided is greater than the balanced section. So the actual neutral axis shift downward i.e., n>nc [Fig. 2.6(d)]. In this section, stress in concrete reaches its permissible value while steel is not fully stressed. Concrete is brittle and it fails by crushing suddenly. As steel is not fully utilised, the over reinforced section is uneconomical (steel is much costlier than concrete). The various features of over reinforced s section are :
(i) Concrete is fully stressed while steel is not (i.e., the stress in concrete is at its permissible value σcbc but stress in steel is less than σst).
(ii) The actual neutral axis is below the critical neutral axis i.e., n > nc.
(iii) The percentage of steel is more than the balanced section, so the section is uneconomical.
(iv) Sudden failure.
(v) The moment of resistance of over-reinforced section is calculated as
\[ M _{r}=\frac{1}{2}\sigma _{cbc}b.n\left ( d-\frac{n}{3} \right )\]
A beam bends under bending moment, resulting in a small curvature. At the outer face (tensile face) of the curvature the concrete experiences tensile stress, while at the inner face (compressive face) it experiences compressive stress.
Definition of beam
Singly reinforced beam
A singly reinforced beam is one in which the concrete element is only reinforced near the tensile face and the reinforcement, called tension steel, is designed to resist the tension.
Doubly reinforced beam
A doubly reinforced beam is one in which besides the tensile reinforcement the concrete element is also reinforced near the compressive face to help the concrete resist compression. The latter reinforcement is called compression steel. When the compression zone of a concrete is inadequate to resist the compressive moment (positive moment), extra reinforcement has to be provided if the architect limits the dimensions of the section.
Under-reinforced beam
An under-reinforced beam is one in which the tension capacity of the tensile reinforcement is smaller than the combined compression capacity of the concrete and the compression steel (under-reinforced at tensile face). When the reinforced concrete element is subject to increasing bending moment, the tension steel yields while the concrete does not reach its ultimate failure condition. As the tension steel yields and stretches, an “under-reinforced” concrete also yields in a ductile manner, exhibiting a large deformation and warning before its ultimate failure. In this case the yield stress of the steel governs the design.
Over-reinforced beam
An over-reinforced beam is one in which the tension capacity of the tension steel is greater than the combined compression capacity of the concrete and the compression steel (over-reinforced at tensile face). So the “over-reinforced concrete” beam fails by crushing of the compressive-zone concrete and before the tension zone steel yields, which does not provide any warning before failure as the failure is instantaneous.
balanced-reinforced beam
A balanced-reinforced beam is one in which both the compressive and tensile zones reach yielding at the same imposed load on the beam, and the concrete will crush and the tensile steel will yield at the same time. This design criterion is however as risky as over-reinforced concrete, because failure is sudden as the concrete crushes at the same time of the tensile steel yields, which gives a very little warning of distress in tension failure.
Steel-reinforced concrete moment-carrying elements should normally be designed to be under-reinforced so that users of the structure will receive warning of impending collapse.
Characteristic strength
The characteristic strength is the strength of a material where less than 5% of the specimen shows lower strength.
Design strength or nominal strength
The design strength or nominal strength is the strength of a material, including a material-safety factor. The value of the safety factor generally ranges from 0.75 to 0.85 in Permissible stress design.