Basis

Mass concrete is a moldable material with good mechanical and durability properties and, although it resists appreciable tensions and compression efforts, it has a very low tensile strength. To adequately resist tensile stresses, it is necessary to combine concrete with a steel skeleton. This skeleton has the mission of resisting the tensile stresses that appear in the structure, while the concrete will resist compression (being cheaper than steel and offering adequate durability properties).

The proper selection of the characteristics of the type of concrete and corrugated steel, in addition to the dimensions and quantities of the materials, allows the collapse of the structural elements to be controlled. For example, an appropriate combination of these variables can allow the steel to yield and thus confer greater ductility so that it deforms enough to be visible before collapse. However, an element with too much corrugated steel inside may present a brittle break, since the bars do not allow that certain ductility and will not show obvious signs of deformation. This is why, in general, the regulations relating to this issue limit the maximum amount of steel that can be placed in the most critical points of the structures.

Longitudinal steel bars, called main or longitudinal reinforcement, are placed in elongated linear elements, such as beams and columns. These steel bars are sized according to the magnitude of the axial force and bending moments, while the shear force and torsional moment determine the characteristics of the transverse or secondary reinforcement.

Calculation of reinforced concrete beams and columns

The Euler-Bernoulli beam theory is not suitable for the calculation of reinforced concrete beams or columns. Resistant elements with these characteristics present a more complex resistance mechanism due to the concurrence of two different materials, concrete and steel, with very different Young's moduli. The moments of inertia are variable according to the size of the cracks in the elements. The different mechanical properties of both materials present in a reinforced concrete element mean that the equilibrium equations that link the internal forces introduced by the forces and tensions of the concrete and the steel are not as simple as those of homogeneous sections, which are those used in the Euler-Bernoulli theory.

In the Spanish Structural Concrete Instruction, mandatory regulations for reinforced concrete structures in Spain until 2021, the mechanical balance equations for the axial force N and the bending moment M of a rectangular section can be written very approximately as:

Where:.

If the normalized rectangle diagram is used to represent the stress-strain relationship of concrete then the tension and compression reinforcement stresses can be expressed in the above functions as:.

On the other hand, the forces supported by the compressed concrete block are given by:.

Section sizing

The problem of section sizing arises when determining the minimum amount of steel to guarantee the adequate resistance of the element. Generally, a reduction in cost is sought, which implies considering various shapes for the section and calculating the reinforcement for each of these possible sections, in order to establish an indicative cost for each section.

A section of a beam subjected to simple bending necessarily requires tension reinforcement (set of bars) placed in the tensioned part of the section, and depending on the bending moment it may also require reinforcement in the compressed part. The area of ​​both reinforcements of a rectangular section can be approximately calculated using the following sets of formulas:.

Where:.

With the same notations, the tensile reinforcement is calculated as:.

Checking sections

The verification problem consists of, given a section completely defined by its geometric dimensions and a certain number of bars with a well-defined arrangement, to verify by calculation whether said section will be capable of supporting the stresses induced in it by the action of known loads.

Basis

Mass concrete is a moldable material with good mechanical and durability properties and, although it resists appreciable tensions and compression efforts, it has a very low tensile strength. To adequately resist tensile stresses, it is necessary to combine concrete with a steel skeleton. This skeleton has the mission of resisting the tensile stresses that appear in the structure, while the concrete will resist compression (being cheaper than steel and offering adequate durability properties).

The proper selection of the characteristics of the type of concrete and corrugated steel, in addition to the dimensions and quantities of the materials, allows the collapse of the structural elements to be controlled. For example, an appropriate combination of these variables can allow the steel to yield and thus confer greater ductility so that it deforms enough to be visible before collapse. However, an element with too much corrugated steel inside may present a brittle break, since the bars do not allow that certain ductility and will not show obvious signs of deformation. This is why, in general, the regulations relating to this issue limit the maximum amount of steel that can be placed in the most critical points of the structures.

Longitudinal steel bars, called main or longitudinal reinforcement, are placed in elongated linear elements, such as beams and columns. These steel bars are sized according to the magnitude of the axial force and bending moments, while the shear force and torsional moment determine the characteristics of the transverse or secondary reinforcement.

Calculation of reinforced concrete beams and columns

The Euler-Bernoulli beam theory is not suitable for the calculation of reinforced concrete beams or columns. Resistant elements with these characteristics present a more complex resistance mechanism due to the concurrence of two different materials, concrete and steel, with very different Young's moduli. The moments of inertia are variable according to the size of the cracks in the elements. The different mechanical properties of both materials present in a reinforced concrete element mean that the equilibrium equations that link the internal forces introduced by the forces and tensions of the concrete and the steel are not as simple as those of homogeneous sections, which are those used in the Euler-Bernoulli theory.

In the Spanish Structural Concrete Instruction, mandatory regulations for reinforced concrete structures in Spain until 2021, the mechanical balance equations for the axial force N and the bending moment M of a rectangular section can be written very approximately as:

Where:.

If the normalized rectangle diagram is used to represent the stress-strain relationship of concrete then the tension and compression reinforcement stresses can be expressed in the above functions as:.

On the other hand, the forces supported by the compressed concrete block are given by:.

Section sizing

The problem of section sizing arises when determining the minimum amount of steel to guarantee the adequate resistance of the element. Generally, a reduction in cost is sought, which implies considering various shapes for the section and calculating the reinforcement for each of these possible sections, in order to establish an indicative cost for each section.

A section of a beam subjected to simple bending necessarily requires tension reinforcement (set of bars) placed in the tensioned part of the section, and depending on the bending moment it may also require reinforcement in the compressed part. The area of ​​both reinforcements of a rectangular section can be approximately calculated using the following sets of formulas:.

Where:.

With the same notations, the tensile reinforcement is calculated as:.

Checking sections

The verification problem consists of, given a section completely defined by its geometric dimensions and a certain number of bars with a well-defined arrangement, to verify by calculation whether said section will be capable of supporting the stresses induced in it by the action of known loads.