Materials¶

The primary materials in reinforced concrete are:

Concrete stress block
Concrete
Reinforcement bars
Mild steel reinforcement bars with definite yield point
High yield strength deformed bars with piece-wise linear stress-strain relation

Concrete Stress Block¶

Concrete stress block repesents the strain distribution and the stress-strain relation so as to be able to obtain strain and stress at a given location as well as obtain the area and first moment of area of the stress distribution which leads to calculation of axial compressive force and moment due to axial compressive force abouth the neutral axis for the case of rectangular sections.

The attributes of a class to represent concrete stress block for limit state method of design are: 1. label: A string representing a label. It will be used as a label in output 2. ecy: Yield strain $\epsilon_{cy}=0.002$, and 3. ecu: Ultimate strain $\epsilon_{cu}=0.0035$

Similarly, a class to repesent the stress block for working stress method of design could be implemented later.

Concrete¶

The attributes of concrete that will be represented are:

label: A string representing a label. It will be used as a label in output
fck: Characteristic strength of concrete in $\text{N/mm}^2$
gamma_m: Partial safety factor for material, which is specified in IS 456:2000 as 1.5
density: Density of concrete in $\text{kN/m}^3$

Reinforcement Bars¶

Rebar is an abstract class representing a steel bar used as longitudinal or shear reinforcement. Reinforcement bars can be of different types, such as mild steel bars and high yield strength deformed (HYSD) bars. Both of them have some common attributes but have significantly different stress-strain relationships. The common attributes that will be available in all child classes are:

label: A string representing a label to refer to the type of reinforcement bar.
fy: Characteristic strength $f_{y}$ of the bar.
gamma_m: Partial safety factor for material, specifid by IS 456:2000 to be $\gamma_m = 1.15$ for all types of reinforcement bars.
density: Density of steel, usually taken to be 78.5 $\text{kN/m}^3$. This will be used to calculate the weight of steel reinforcement.
Es: Modulus of elasticity for steel, specifid by IS 456:2000 to be $E_s = 2 \times 10^5$ for all types of reinforcement bars.

Reinforcement bars are specified as having a linear stress-strain relationship up to a certain point and stress is assumed to remain constant beyond yield up to infinity. While no material is truly infinitely ductile, if the ductile portion is sufficiently long in relation so as to permit concrete to reach its ultimate strain, it would be justified to call it infinitely ductile.

Mild Steel Reinforcement Bars¶

Mild steel reinforcement bars, represented by the RebarMS class, are known to have a well defined yield point and therefore have a distinct bilinear stress-strain relationship. This class does not need any additonal attributes beyond what is already defined in the parent Rebar class. It differs from HYSD reinforcement bars only in the stress-strain relationship. The stress is linear up to the yield stress and after yield, the stress is constant. Yield stress is $\frac{f_y}{\gamma_m} = \frac{f_y}{1.15} = 0.87 f_y$ and yield strain is $\frac{1}{E_s} \frac{f_y}{\gamma_m} \approx \frac{0.87 f_y}{2 \times 10^5}$.

High Yield Strength Deformed Bars¶

High Yield Strength Deformed (HYSD) reinforcement bars, repesented by the RebarHYSD class, have a higher yield strength as implied by their name and differ from mild steel bars in that they have a piece-wise linear stress-strain relationship. ReabHYSD is derived from the abstract Rebar class. As per the LSM of IS 456:2000, the stress-strain relation between the design stress of $0.8 \frac{f_y}{\gamma_m}$ and $\frac{f_y}{\gamma_m}$ is piece-wise linear and inelastic strain for each of the stress levels between 0.8 and 1.0 times $\frac{f_y}{\gamma_m}$, namely, 0.85, 0.9, 0.95 and 9.975 are specified as 0.0001, 0.0003, 0.0007, 0.001 and 0.002 respectively. From this information, it is possible to arrive at the complete stress-strain relation and determine the stress corresponding to a given strain. The additional information required is the table of inelastic strains for the different levels of design stress calculated based on the specified inelastic strain for different stress levels.

Layers of Reinforcement Bars¶

Reinforcement bars are placed in layers parallel to one of the principal axes of a section. The atrributes of a layer of reinforcement bars are:

rebar: An object of type Rebar representing the type of reinforcement bar and its properties. Thus it is presumed that all bars in a layer are of the same type.
dia: List of diameter of bars in the layer. It is assumed that all bars in a layer are of the same type.
_dc: Distance of the centroid of the layer from one of the edged of the section. If the distance is positive, it is interpreted as the distance from the highly compressed edge. If the distance is negative, it is interpreted as the distance from the tension edge. This assumes that the section has an edge parallel to one of the principal axes.
_xc: Distance of the centroid of the layer from the highly compressed edge. This value is calculated once the size of the section is known. It is not an input.
stress_type: A string, 'C' if the layer is in compression and 'T' if it is tension. It is not an input, but will be decided during analysis of the section depending on where the layer is located with respect to the neutral axis.

Groups of Layers of Reinforcement Bars¶

Layers of reinforcement bars in a section are arranged in groups. For example, one layer near the compression edge and perhaps one or more near the tension edge. All layers are considered as part of a group of layers of reinforcement bars. Whether a particular layer of reinforcement bars within a group is subjected to tension or compression will only be known when the location of the neutral axis is computed. This is especially true of layers located close to the actual location of the neutral axis that is yet to be computed.

Attributes of a group of reinforcement bars are:

layers: List of objects of type RebarLayer. A group may have zero or more layers of reinforcement bars.

Shear Reinforcement¶

Concrete, without any tension reinforcement has an ability to resist shear. However, when the shear force on a section exceeds the shear capacity of concrete, it will be necessary to either increase the depth of the cross section, if that is an option, or to provide shear reinforcement to resist the additional shear beyond the capacity of concrete if depth cannot be increased. Shear reinforcement can be provided in one of several ways:

Stirrups, either vertical or inclined.
Single group of parallel bars bent-up at an angle at a single location.
A series of groups of parallel bars bent-up at an angle at a regular spacing. While this is possible, it is not commonly used.

However, there is an upper limit to the shear that can be resisted by a section with shear reinforcement. If the shear force on a section exceeds this limit, the only solution is to increase the depth of the cross section.

The ShearReinforcement is an abstarct class. The attributes of shear reinforcement are:

rebar: The type of reinforcement bars used as shear reinforcement.
_Asv: Area of shear reinforcement. This is not an input data. It will be computed based on the type of reinforcement.
_sv: Spacing of shear reinforcement. This is an input data and is common for all types of shear reinforcement.

Vertical and Inclined Stirrups¶

Reinforcement bars bent into the shape of a link and provided at regular spacing can be used to resist shear force at a section. The Stirrups class is a child class of ShearReinforcement. While vertical bars make an angle of 90 degrees with the longitudinal axis of a beam, inclined stirrups usually make an angle of 45 degrees. However, the angle of inclination can be other than 45 degrees. The shear resisted by a stirrup depends on the angle of the stirrup and increases as he angle approaches 90 degrees. The attributes of this type of shear reinforcement are:

_nlegs: Number of legs of the stirrup.
_bar_dia: Diameter of reinforcement bar used as stirrup.
_alpha_deg: Angle in degree made by the stirrup with the longitudinal axis of the beam. It is 90 degree for vertical stirrups.

Bent-up Bars¶

A single group of parallel bent-up bars used as shear reinforcement are the longitudinal tension bars that are bent up, usually at 45 degrees. Not all tension bars at midspan are required at the ends of a simply supported beam where the bending moment is zero. In the case of a fixed or continuous beam, tension is at the top edge at the supports and bottom bars are not required, unless they are used as compression bars. In any case, some bottom bars at midspan can be used as bent-up bars to augment shear capacity or can be curtailed otherwise. It must be kept in mind that bent up bars are effective over a limited length of the beam. A series of groups of parallel bent-up bars act similar to inclined stirrups. If _sv is zero, the bent-up bars will be treated as a single group of bent-up bars and if _sv is non-zero, it will be treated as a series of groups of parallel bent-up bars.

The attributes of bent-up bars as shear reinforcement, in addition to _sv, are:

bars: List of diameters of bars that are bent-up. This can be subset of a layer of reinforcement that will be bent up.
_alpha_deg: Angle in degrees made by the bent-up bars with the longitudinal axis of the beam.

Group of Shear Reinforcements¶

A section may be provided with more than one type of shear reinforcement. For example, if the design shear force is large resulting in very close spacing of vertical stirrups, a single group of parallel bent-up bars close to the section of maximum shear may help in increasing the spacing of vertical stirrups. Such an arrangement of shear reinforcement is represented by ShearRebarGroup, which is a list of one or more types of shear reinforcements. While it is theoretically possible to use any combination of shear reinforcement types, it is common to use vertical stirrups sometimes augmented by single group of bent-up bars.

The only attribute of ShearRebarGroup is:

shear_reinforcement, which is a list of one or more shear reinforcement types, such as vertical stirrups, inclined stirrups, single group of bent-up bars or a series of bent-up bars.

LateralTie¶

Lateral ties in columns help to confine concrete in a column section and play a critical role in enhacing ductility of column sections. They are not expected to carry any stress. The attributes of a lateral tie are: 1. rebar: Object of type Rebar repersenting the reinforcement bar used as the lateral tie 2. bar_dia: Diameter of the lateral tie 3. spacing: Spacing between lateral ties