## Curvature

Curvature is defined as change of angle per unit length along the axis of a object under the action of bending loads at any given section.

## Moment

Moment (force) is a magnitude of tendency to cause an object to rotate with respect to a specific axis or point under the action of a force. Force is included here as it is related to the derivation of this relationship; moment may be of other physical quantity like charge, mass etc.To produce any significant value of moment, the force which result rotation, must be placed in such a way that this will initiate twist in the body. This phenomenon only happened when line of action of force is not colinear with the centroid of respective body.

The magnitude of moment of force about an axis or point has direct relation to the distance between line of action of force and axis about which it rotates. This distance is called moment arm. Moment arm (often called lever arm) is measured orthogonally between line of action of force and center of moment. If force is applied obliquely, only perpendicular component will produce moment.

Moment= Force X moment arm

M= F X d

The magnitude of moment of force about an axis or point has direct relation to the distance between line of action of force and axis about which it rotates. This distance is called moment arm. Moment arm (often called lever arm) is measured orthogonally between line of action of force and center of moment. If force is applied obliquely, only perpendicular component will produce moment.

Moment= Force X moment arm

M= F X d

## Bending moment

When external force (s) or moment (s) is/are applied on an element resulting it to bend, as a reaction bending moment is induced. Bending moment at any section of a structural element can be defined as the summation of moments about this section produced by all external forces that act on one side (either right or left) of this section.

The internal reactions of cross-section of a structural element can be taken as combination of resultant force and resultant couple. To have equilibrium, moment produced by external loading (force and moment) must be counteracted by couple provided by internal loads. The result of internal forces is termed as shear force or normal force (depending on plane of action) and resultant of internal couple is termed as bending moment.

M=∑M

_{L}=∑M_{R }
M

_{L}stand for bending moment calculated based on loads that act on left side of the section and M_{R }stand for that at right side of section.
Positive bending moments will produce bending of a beam to concave upward i.e. beam curves downward at the middle (known as sagging); Whereas negative bending moment will result hogging of a beam i.e beam curves upward at the middle.

Modulus of rupture is very common term in concrete engineering. It is also known as flexural strength or transverse rupture strength (f

### Modulus of rupture

Modulus of rupture is very common term in concrete engineering. It is also known as flexural strength or transverse rupture strength (f

_{r}). It is the stress just before yielding of a material in flexure test. It represents the maximum stress that a material can experience at the moment of yielding; so the unit of stress is the measuring unit of modulus of rupture.
A homogeneous object (i.e. object consist of single materials) like steel rod or wooden beam will subject to variable stresses throughout its depth under normal service loading. At the edge of concave face (inside of bend) of an object, the stress will reach maximum value and its nature will be compressive. At the edge of convex face (outside of bend), the stress will reach maximum value and is of tensile stress. The inner most and outermost edge of a beam where maximum stresses are induced are termed as extreme fiber. Most of the materials like concrete, it is observed to suffer tensile failure; so the maximum value of stress that can withstand before its failure is known as flexural strength or modulus of rupture of that object.

Flexural stress can be determined by σ = My/I ..(1)

When stress of extreme fiber need to be determined y is replaced by c where C is distance between neutral axis and remotest fiber.

σ = MC/I ..(2)

Flexural stress can be determined by σ = My/I ..(1)

When stress of extreme fiber need to be determined y is replaced by c where C is distance between neutral axis and remotest fiber.

σ = MC/I ..(2)

Equation (1) and (2) can be used to determine flexure stress of a beam under rupture in testing machine. As at that stage proportional limit of material is exceeded, the stress found in this equation is not the actual stress; but the imaginary stress obtained with this method is known as modulus of rupture. This property of material is used to make comparison of ultimate strength of flexure members of various materials and sizes.

### Bending stress and shear stress:

Internal forces acting on any cross section can be resolved into two components; tangential and normal to that section. Of these components, that act normal to section are called bending stresses which will produce compression on one side and tension on other side. The function of bending stress is to counteract bending moment.

The components that are tangential to the section are called shear stresses, the functions of which are to resist shear or transverse forces.

### Transformed concrete section

When stress in concrete section is low (i.e. ≤f’

_{c}/2), concrete is found to act more or less elastically; this means stress is nearly proportional to strains Figure-2 shows the line d which represents this behavior with small error under both slow and fast loading. At this stress, normal weight concrete shows strain of the order of about 0.0005, whereas steel behaves elastically nearly at its yield points (60 ksi) which may be represented by a strain 0.002 (much greater than concrete).Figure-2: Stress-strain curve of concrete and steel |

Form the assumption of prefect bond, as compression strain in concrete is equal to compression strain in steel at a given load,

Reinforced concrete members are non-homogeneous as they consist of two completely different materials; so the procedures to analyze reinforced concrete members are not the same as that of methods used in analyzing or designing of members of homogeneous materials like wood, steel or other materials. The basic principles considered, though, essentially identical.

Significance of moment-curvature relationship:

Although this relationship is not included in ACI code and also is not required evidently in general design, the curvature resulted from moment put on a particular section of beam under full extent of loading leading to failure is required in different contexts. This relationship is the base for studying

• Ductility of member

• Understanding the formation of plastic hinge

• Calculating redistribution of elastic moments which may occur in reinforced concrete members before collapse.

Reinforced concrete members are non-homogeneous as they consist of two completely different materials; so the procedures to analyze reinforced concrete members are not the same as that of methods used in analyzing or designing of members of homogeneous materials like wood, steel or other materials. The basic principles considered, though, essentially identical.

Significance of moment-curvature relationship:

Although this relationship is not included in ACI code and also is not required evidently in general design, the curvature resulted from moment put on a particular section of beam under full extent of loading leading to failure is required in different contexts. This relationship is the base for studying

• Ductility of member

• Understanding the formation of plastic hinge

• Calculating redistribution of elastic moments which may occur in reinforced concrete members before collapse.

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