Slope and deflection of beam by direct integration method (Macaulay's Method)

Integration method

(a) Direct Integration; This method is based on the differential equation of elastic curve. Integrating once the above equation of elastic curve gives an equation in terms of dv/dx with an unknown constant c1as follows;

EI (dv/dx) = Mx +c1

the value of constant c1 can be determined by applying the conditions of slope and/or deflection at some known points of the beam.

Similarly, integration of the equation again will give an equation in terms of "v" with an unknown constant c2 as given below;

EI v = Mx2/2 + c1x + c2

The value of constant c2 can be determined by applying the conditions of deflection at some known points.

(b) Macaulay's method; This method is modified form of integration method and used when the loading is discontinuous and/or discrete. In other words we can say that a single equation is not sufficient to find bending moment along the span of the beam. In this method we put all the terms containing x into square bracket and we integrate the bracket not "x". During calculation for the values of slope and deflection, any bracket with negative value is to be ignored. An equivalent loading is required to be applied in case of uniformly distributed load. See Example 6-1 and  Example 6-2 for more explanations.

You can use our Deflection Calculator Easy to use calculator for different loads on beams

#### Excellent Calculators

Stress Transformation Calculator
Calculate Principal Stress, Maximum shear stress and the their planes

Calculator for Moving Load Analysis
To determine Absolute Max. B.M. due to moving loads.

Bending Moment Calculator
Calculate bending moment & shear force for simply supported beam

Moment of Inertia Calculator
Calculate moment of inertia of plane sections e.g. channel, angle, tee etc.

Reinforced Concrete Calculator
Calculate the strength of Reinforced concrete beam

Moment Distribution Calculator
Solving indeterminate beams

Deflection & Slope Calculator
Calculate deflection and slope of simply supported beam for many load cases

Fixed Beam Calculator
Calculation tool for beanding moment and shear force for Fixed Beam for many load cases

BM & SF Calculator for Cantilever
Calculate SF & BM for Cantilever

Deflection & Slope Calculator for Cantilever
For many load cases of Cantilever

Overhanging beam calculator
For SF & BM of many load cases of overhanging beam

Civil Engineering Quiz
Test your knowledge on different topics of Civil Engineering

Research Papers
Research Papers, Thesis and Dissertation

List of skyscrapers of the world
Containing Tall building worldwide

Forthcoming conferences
Containing List of civil engineering conferences, seminar and workshops

Profile of Civil Engineers
Get to know about distinguished Civil Engineers

Professional Societies
Worldwide Civil Engineers Professional Societies

Keep visitng for getting updated or Join our mailing list

#### Search our website for more...

Please Tell your Friends about us if you find our website useful