**Method of Virtual Work (Unit Load Method) for slope and deflection**

Unit load method also referred to as method of virtual work was developed by John Bernoulli in 1717. It is used to determine the slope and deflection at a specified point of a beam, frame, or truss.

In this method a virtual unit load is applied at the point of deflection (δ) and in the direction of deflection, whereas a unit virtual moment is applied at the point of desired slope.

By applying the principle of work and energy we get

external virtual work = internal virtual work

**External virtual work** =1kN.δ

Therefore, δ **= **Internal virtual work

Based on internal virtual work for different types of loading the deflection formula is given as follows;

__Flexural members__

In case of flexural members (beam, frame) the total internal virtual work is obtained by integration.

Where δ = deflection at point C

M_{x} =
bending moment at x due to actual loading

m_{x} =
bending moment at x due to virtual unit load applied at point of
deflection, C.

L = beam span

E = Modulus of elasticity of beam material

I = moment of inertia of beam section.

Note: *In case of calculation of
slope at a point we should apply a unit moment at the
specified point of slope and write the equation for
m _{x} *

See our solved Problem 7-3 for deflection of beam by unit load method.

__Pin-jointed
Truss__

In case of truss the total internal virtual work is obtained by taking the algebraic sum

Where,

δ_{C}
= deflection of joint C

N=axial force in a member due to real loading

n = axial force in the member due to virtual unit load applied at the joint C

L= length of member

A= area of cross-section of the member

E = modulus of elasticity of the member

Note*: compressive
forces are taken as negative whereas tensile forces are taken as
positive.*

See our solved Problem 7-5 for deflection of Truss by unit load method

For other problems please visit our
**
Problem Solver**

You can also use our
**
Moment Distribution Calculator**
for solving indeterminate beams

Use our
**
Fixed Beam
Calculator **to
determine fixed-end moments for different loading cases.

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 visiting for getting updated or Join our mailing list

PleaseTell your Friends about us if you find our website useful