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**2E4: SOLIDS & STRUCTURES Lecture 8**

Dr. Bidisha Ghosh Notes: lids & Structures

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**Properties of Sections**

Cross-sections of beams: Cross-Sections of other structural or machine elements: To find out stress or deformation we need to know about the geometric properties of these sections!

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Centroid Centroid is the geometric centre which represents a point in the plane about which the area of the cross-section is equally distributed. Centre of gravity for a body is a point which locates the gravity or weight of the body. Centroid and CG are same for homogeneous material.

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**Moment of Area This is called the First Moment of Area**

Take a infinitesimally small area (dA) in the shaded area (area under consideration). Moment of this area about the point O, Moment of the entire shaded area about the point O can be by summing over all such small dA areas or by, First moment of area about x-axis or y-axis, This is called the First Moment of Area An important concept to find out centroid. The limits of the integration are decided based on the dimensions (end points) of the area under consideration.

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**Calculating position of centroid**

The centroid of the entire shaded area (set of areas dA) is the point C with respect to which the sum of the ﬁrst moments of the dA areas is equal to zero. The centroid is the point deﬁning the geometric center of system or of an object.

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Centroid of a Triangle

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Composite areas When a composite area is considered as an assemblage of n elementary areas, the resultant moment about any axis is the algebraic sum of the moments of the component areas. Therefore the centroid of a composite area is located by,

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**Centroid of an L-Shaped Area**

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**Centroid of an L-Shaped Area**

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**Can you write a matlab/excel code to calculate moment of inertia?**

Moments of Inertia MOI is a measure of the resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics. It is the second moment of area, Radius of gyration, (the distance at which the entire area can be assumed to be distributed for calculation of MOI) Can you write a matlab/excel code to calculate moment of inertia?

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**Moments of Inertia of a rectangle**

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**Polar Moment of Inertia**

This is the moment of inertia of a plane area about an axis perpendicular to the area.

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Parallel Axis Theorem The parallel-axis theorem relates the moment of inertia of an area with respect to any axis to the moment of inertia around a parallel axis through the centroid.

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**Moment of Inertia of an I-beam**

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