Section Modulus calculation and optimization

Section modulus :

                         It is a geometric property for a given cross-section used in the design of beams or flexural members. Other geometric properties used in design include area for tension and shear, radius of gyration for compression, and second moment of area and polar second moment of area for stiffness. Any relationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapes are given below. There are two types of section moduli, the elastic section modulus and the plastic section modulus. The section moduli of different profiles can also be found as numerical values for common profiles in tables listing properties of such.

Notation

North American and British/Australian convention reverse the usage of S & Z. Elastic modulus is S in North America, but Z in Britain/Australia,and vice versa for the plastic modulus.Eurocode 3 (EN 1993 - Steel Design) resolves this by using W for both, but distinguishes between them by the use of subscripts - W_el and W_pl.

Elastic section modulus

For general design, the elastic section modulus is used, applicable up to the yield point for most metals and other common materials.

The elastic section modulus is defined as S = I / y, where I is the second moment of area (or area moment of inertia, not to be confused with moment of inertia) and y is the distance from the neutral axis to any given fibre. It is often reported using y = c, where c is the distance from the neutral axis to the most extreme fibre, as seen in the table below. It is also often used to determine the yield moment (M_y) such that M_y = S ⋅ σ_y, where σ_y is the yield strength  of the material.

Section modulus equations

Cross-sectional shape	Figure	Equation	Comment
Rectangle		�=�ℎ26	Solid arrow represents neutral axis
doubly symmetric I-section (major axis)		��=��26−�ℎ36� ��=���, with �=�2	NA indicates neutral axis
doubly symmetric I-section (minor axis)		��=�2(�−ℎ)6+(�−�)3ℎ6�	NA indicates  neutral axis
Circle		�=��332	Solid arrow represents neutral axis
Circular hollow section		�=�(�24−�14)4�2=�(�24−�14)32�2	Solid arrow represents neutral axis
Rectangular hollow section		�=��26−�ℎ36�	NA indicates neutral axis
Diamond		�=��224	NA indicates neutral axis
C-channel		�=��26−�ℎ36�	NA indicates neutral axis

Plastic section modulus

The plastic section modulus is used for materials where elastic yielding is acceptable and plastic behavior is assumed to be an acceptable limit. Designs generally strive to ultimately remain below the plastic limit to avoid permanent deformations, often comparing the plastic capacity against amplified forces or stresses.

The plastic section modulus depends on the location of the plastic neutral axis (PNA).The PNA is defined as the axis that splits the cross section such that the compression force from the area in compression equals the tension force from the area in tension. So, for sections with constant, and equal compressive and tensile yielding stress, the area above and below the PNA will be equal, but for composite sections, this is not necessarily the case.

The plastic section modulus is the sum of the areas of the cross section on each side of the PNA (which may or may not be equal) multiplied by the distance from the local centroids of the two areas to the PNA:

��=����+����

The Plastic Section Modulus is not the 'First moment of area'. Both relate to the calculation of the centroid, but Plastic Section Modulus is the Sum of all areas on both sides of PNA (Plastic Neutral Axis) and multiplied with the distances from the centroid of the corresponding areas to the centroid of the cross section, while the First moment of area is calculated based on either side of the "considering point" of the cross section and it is different along the cross section and depends on the point of consideration.

Uniformly distributed load

Let us consider the following figure, a beam AB of length L is loaded with uniformly distributed load and rate of loading is w (N/m).

Total uniformly distributed load, P = w*L

 

 

Uniformly varying load

Uniformly varying load is the load which will be distributed over the length of the beam in such a way that rate of loading will not be uniform but also vary from point to point throughout the distribution length of the beam.

 

Uniformly varying load is also termed as triangular load. Let us see the following figure, a beam AB of length L is loaded with uniformly varying load.  

We can see from figure that load is zero at one end and increases uniformly to the other end. During determination of the total load, we will determine the area of the triangle and the result i.e. area of the triangle will be total load and this total load will be assumed to act at the C.G of the triangle.

Total load, P = w*L/2

 

 

 

 

 

 FIRST INTERSECTION :

                                                                      Z=I/Y

                                                                      I=3.232e^-007m^4

                                                                      Y=448.445mm

                                                              Z=3.232e^-007m4/448.445mm

                                                              Z=6.661159103*10^-3m^3

SECOND INTERSECTION :

                                                                      Z=I/Y                 

                                                                     I=3.719e^-007m4

                                                                     Y=448.445mm

                                                                 Z=3.719e^-007m4/448.445mm

                                                                   Z= 7.562301759*10^-3m^3