Slope and Deflection of a beam

Whenever a beam is loaded with transverse loads, the bending moments are developed which cause the axis of beam to deflect from the original undisturbed position as seen in the following figure.

slope and deflection

In the above figure the undisturbed axis of cantilever beam is AB and the deflected shape is AB'. The deflected shape of the beam is also known as elastic curve.  The deviation of point B to B' is shown as deflection  δB and the change in slope of tangent at B is shown as slope θB .

The differential equation of elastic curve was first given by Euler and written as

          elastic curve equation

where   E = Modulus of elasticity of the beam material

             I = moment of inertia of the beam x-section

             v = deflection along y-axis

             x = distance from the origin to the point of deflection

             M = bending moment at section

Deflection calculations are required for buildings, bridges and machines to satisfy some design criteria and to control vibration.

There are different methods available to determine slope and deflection of beam. Some of them are given below;

  1. Double Integration method/Macaulay's Method
  2. Moment-Area method
  3. Conjugate-beam method
  4. Strain-energy method (Castigliano's theorems)
  5. Virtual work method (Unit Load method)

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

Moment Distribution CalculatorNewfor solving Indeterminate Beams with different loads

You can also select from the following links of solved examples to get prepared for your exams like GATE and GRE.

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

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