MIDDLE THIRD RULE FOR RECTANGULAR SECTION

In our previous topics, we have seen some important concepts such as Concepts of direct and bending stresses, bending stress in beams, basic concept of shear force and bending moment, strain energy stored in body, beam bending equation, bending stress of composite beam, shear stress distribution diagram for various sections etc.

Now we will be concentrated here a very important topic i.e. Middle third rule for rectangular section with the help of this post.

Cement concrete columns are weak under tensile load and therefore we must be sure that there should not be any tensile load anywhere in the section and hence load must be applied in such a way that there will be no tensile stress developed in the section of cement concrete columns.

As we have discussed that when a body will be subjected with an axial tensile or axial compressive load, there will be produced only direct stress in the body. Similarly, when a body will be subjected to a bending moment there will be produced only bending stress in the body.

Now let us think that a body is subjected to axial tensile or compressive loads and also to bending moments, in this situation there will be produced direct stress and bending stress in the body.

If a column will be subjected with an eccentric load then there will be developed direct stresses and bending stresses too in the column and we will determine the resultant stress developed at any point in the column by adding direct and bending stresses algebraically.

Principle used

We will consider here compressive stress as positive and tensile stress as negative and we will have the value of resultant stress at any point in the column section. There will be maximum stress and minimum stress in the section of column as mentioned here.

σ_Max = Direct stress + Bending stress

σ_Max = σ_d+ σ_b

σ_Min = Direct stress - Bending stress

σ_Min = σ_d- σ_b

If minimum stress σ_Min = 0, it indicates that there will be no stress at the respective point in the section

If minimum stress σ_Min = Negative, it indicates that there will be tensile stress at the respective point in the section

If minimum stress σ_Min = Positive, it indicates that there will be compressive stress at the respective point in the section

Let us come to the main subject that is Middle third rule for rectangular section

Let us consider a rectangular section of area A and of width b and Depth d as displayed in following figure. Let us consider that an eccentric load P is acting over the rectangular section with eccentricity e with respect to axis YY.

Minimum stress at any point in the section will be given by following formula as mentioned here

As we have seen above the various conditions of minimum stress values and their importance and therefore we can easily say that minimum stress (σ_Min) must be greater or equal to zero for no tensile stress at any point along the width of the column.

Let us analyze the above equation and we will conclude that in order to not develop any tensile stress at any point in the section along the width of the column, eccentricity of the load must be less than or equal to (b/6) with respect to axis YY.

Therefore we can say that if load will be applied with an eccentricity equal to or less than b/6 from the axis YY and on any side of the axis YY then there will not be any tensile stress developed in the column.

Hence range within which load could be applied without developing any tensile stress at any point of the section along the width of the column will be b/3 or middle third of the base.

Similarly in order to not develop any tensile stress at any point in the section along the depth of the column, eccentricity of the load must be less than or equal to (d/6) with respect to axis XX.

Therefore we can say that if load will be applied with an eccentricity equal to or less than d/6 from the axis XX and on any side of the axis XX then there will not be any tensile stress developed in the column.

Hence range within which load could be applied without developing any tensile stress at any point of the section along the depth of the column will be d/3 or middle third of the depth.

Now let us consider that load is eccentric with respect to axis XX and axis YY both, in this situation load must be applied anywhere within the rhombus ABCD whose diagonals AC = b/3 and BD= d/3 in order to not develop any tensile stress at any point in the column.

Area ABCD within which load could be applied without developing any tensile stress at any point of the section will be termed as Kernel of the section.

We will discuss Middle quarter rule for circular section in our next post.

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