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 = Mx^{2}/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

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

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

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