Moment Of Inertia Crack Section Transformed To Concrete

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 m oment, 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 fo rce is not c olinear 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. Th is 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 for ce is applied obliquely, only per pendicular component will produce moment.Moment= Force X moment armM= F X dBending moment. 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 ruptureModulus 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.

Moment Of Inertia Cracked Section Transformed To Concrete Walls

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 byσ = 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 = M C/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.

Moment Of Inertia Cracked Section Transformed To Concrete Stone

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. 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.

Plain concrete member has insufficient flexural strength as tensile strength as tensile strength of it in bending (f r) is a mere fraction of strength in compression. As a result, this beams fail at the tension side under smaller loads before concrete at the compression side is stressed to failure.

This problem is solved by introducing reinforcing steel bars at tension face of member as near as possible to the extreme fiber. Thus clear cover only left outside the reinforcing bar to ensure protection against corrosion and fire.Thus in reinforced concrete flexural member like beam, tension resulted from beam bending moments is mainly resisted by the reinforcing bars whereas concrete is considered capable to resist corresponding compression. This combined action is valid so far these two distinct materials are not subjected to relative slip past another.Thus bond is the key to behave such composite member to behave as a unit which is achieved by applying deformed bars as reinforcement. Deformed bars improve bond strength by mechanical bond along with chemical bond at concrete-steel interface.

When bond stress exceeds bond strength, necessary anchorage is provided at the ends of reinforcing bars. Though different shapes of concrete masses can be produced, for simplicity of discussion a beams with rectangular cross-section will be considered.When load is applied on such a beam gradually, stress vary from zero to the value at which beam will fail, the distinguished behavior of the beam at different stages are observed.At earlier stage of loading when tensile stress induced in concrete is lower than modulus of rupture, the entire section is effective to resist stress irrespective of position with respect to neutral axis (NA)i.e. Effective in compression face and effective at tension face located on the other side of neutral axis. At this stage the reinforcement deforms at the same amount as that of concrete and also subjected to tensile stress.

At this level of loading, stress in concrete are small and is proportional to strain. The distribution of stresses and strains in steel and concrete throughout the depth of section are shown figure-5.Figure-5:Stress-strain distribution of reinforced concrete beam (f ct.

Figure 9: Generation of tension cracksWhen tensile stress in concrete is more than f r, tension cracks are generated as shown in figure 9. At compression stress on concrete is less than about 1/2f’ c, stress in steel remains below yield point; thus both materials act elastically or close to elastic. This is the situation of structure in normal service loads and conditions.In this situation, it is assumed that tension cracks will propagate toward the neutral axis; for simplicity of calculation with little error, concrete that is subjected to tensile stress is cracked, thus it is considered not effective. Thus the transformed section is still valid for analysis except concrete under neutral axis (concrete at tension side) is deleted.Figure 10: Stress-strain distribution of cracked beam under elastic loadingFigure-10 shows the transformed section of cracked concrete beam which consists of concrete on the compression side of neutral axis and n time of total steel area on the other side. For calculation we need to determine location of neutral axis.

Moment Of Inertia Cracked Section Transformed To Concrete Floor

The distance of neutral axis from the top fiber are taken as s fraction of effective depth; kdMoment about tension areas about neutral axis= moment that of compression area. This will yield point 3 in moment-curvature graph. The curvature of point 2 can be easily determined by the ratio ofStress inelastic, section crackedIt is evident that at or close to ultimate load, the stress-strain relationship is no longer proportional. In case of axial compression and bending, it was found that at higher load near failure, the distribution of stress strain are not of elastic distribution rather they are that of Figure-11.

Geometrical shape of distribution of stress varies based on several factors like duration and rate of loading and cylinder strength.Figure 11: Stress-strain distribution of cracked beam under inelastic loading. Now concrete section is subjected to stress in inelastic range, though steel is not yield yet. Interesting thing that neutral axis lies above its elastic location i.e.