# How to use the Double integration method for an overhanging beam?

• Guruprakash
In summary, the double integration method is used to calculate bending moments in a beam that is not statically determinate. It involves integrating the shear force equation twice to get the moment equation. In this method, the influence of a point load is accounted for by the reaction forces, which can be calculated from the point load. The equation for an overhanging beam with an offset point load at a specific section is correct and includes all the terms for reactions and the point load. When taking a section from right to left, there will be no reaction term in the equation. However, the use of double integration method is not necessary for calculating bending moments in statically determinate beams, as the equilibrium equations can be used instead.
Guruprakash
TL;DR Summary
Forming an equation for beam with point load at the end is well known. When we take a section at distance 'x' we ought to neglect the point load or reaction at end of the beam. But how the force influence is take into account in the equation? But when we form a equation for overhanging beam with point load which is offset from the end we include all the force and reaction in the equation.
In case of overhanging beam with point load at the end. For example:
(here RA-reaction is negative)

The equation will be as follows (by double integration method):
, as we can see the equation will not have Point load (10kN) term in it.

1) How the influence of the point load is accounted in this equation? Is it accounted by the reaction forces ([R][/A], [R][/B]) as it is calculated from the point load(10kN)?

In case of overhanging beam with offset point load from the end. For example:

The equation is (for section at x=12)
.

2) Is the above equation is correct? If yes, it has all the terms of reactions and the point load and if we take a section from right to left there will be no reaction (RA) term in the equation.

Kindly explain me how double integration method equation works.

You don’t have to use double integration if you are just looking for bending moments in a statically determinate beam. The moments at any section along the beam are determined by the equilibrium equations. In particular, the sum of moments about any point on the beam must equal zero, and that includes summing moments about a load or support point.

## 1. What is the Double Integration Method for an overhanging beam?

The Double Integration Method is a mathematical technique used to determine the deflection and slope of an overhanging beam under various loading conditions. It involves solving two differential equations, one for the deflection and one for the slope, by integrating the bending moment equation twice.

## 2. When is the Double Integration Method used for an overhanging beam?

The Double Integration Method is typically used when the overhanging beam has a continuous load distribution, such as a uniformly distributed load, and the support conditions are not simple (e.g. fixed or pinned). It is also useful for beams with varying cross-sections or material properties.

## 3. What are the advantages of using the Double Integration Method?

The Double Integration Method allows for a more accurate calculation of deflection and slope compared to other methods, such as the Moment-Area Method. It also takes into account the effects of shear and bending moments, making it suitable for more complex beam problems.

## 4. What are the limitations of the Double Integration Method?

The Double Integration Method can be time-consuming and requires a good understanding of differential equations. It also assumes linear elastic behavior of the beam material and neglects the effects of shear deformation, which may lead to slightly inaccurate results for highly flexible beams.

## 5. Are there any tips for using the Double Integration Method effectively?

It is important to carefully define the coordinate system and sign conventions for the deflection and slope equations. It is also helpful to break the beam into smaller segments and solve for the deflection and slope at each segment separately before combining the solutions. Additionally, double check all calculations and be mindful of units to avoid errors in the final result.

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