The principle of virtual work sometimes known as the “unit-load” method was developed by John Bernoulli in 1717. It is a direct application of the principle of conservation of energy which states that the work by all external forces acting on a structure, W_{e}, is transformed into internal work or strain energy, W_{i} developed as a result of the structure’s deformation under these forces. Supposing these forces are removed. the strain energy stored will return the structure to it un-deformed state-provided the elastic limit has not been exceeded. Thus, energy is conserved.

The principle can be mathematically expressed as :

{ W }_{ e }={ W }_{ i }

Where:

- W
_{e}is the external work done by the external forces - W
_{i}is the internal work done or strain energy stored in the structure

The concept of virtual work can be extended further by considering a structure subjected to a series of external forces P_{1}, P_{2}, P_{3}……P_{n}. Supposing it is required to determine the displacement of the structure at a point A (figure 1) due to the applied external forces. Since there is no external force applied at point A, the displacement of point A can be determined by applying a “virtual force” at point A acting in the same direction of the displacement. Its value is often taken to have a magnitude of “unity”. The term virtual load is chosen to depict an imaginary load that doesn’t exist as part of the real loads. However, the unit load does cause internal forces in the member as shown in figure 1. We can, therefore, apply the principle of conservation of energy described above to determine this displacement.

Once the unit virtual force P’ is applied, the structure is subjected to real loads and point A will displaced an amount Δ causing the structure to also displace internally by dL. Thus the displacement caused by the unit virtual force P’ is Δ and that caused by the internal forces u is dL. Therefore applying the principle of conservation of energy, we have:

P'.\Delta =u.dL

Where:

- P’ = 1=Unit Virtual Force acting in the direction of Δ
- u = virtual internal forces acting in the direction of dL
- Δ = external deflection caused by the real loads
- dL = internal deformation of the element caused by the real loads

By choosing P’ as “unity” we can see that the solution of Δ

**Application of Virtual Work to Trusses**

The principle of virtual work can be applied to determine the displacement of trusses subjected to external forces. Supposing we intend to determine the displacement of a joint in a truss, the virtual work equation for a truss can be expressed as:

1\cdot \Delta =\sum { \frac { nNL }{ AE } }

Where :

- 1= external virtual unit load acting on the truss joint in the stated direction of Δ
- n = internal virtual normal force in a truss member caused by the external virtual unit load
- Δ = external joint displacement caused by the real loads on the truss
- N = internal normal force in a truss member caused by the real loads
- L = length of a member
- A = cross-sectional area of a member
- E = modulus of elasticity of a member.

**Procedure for Analysis**

The following steps may be applied to obtain the displacement at any joint in the truss using the virtual work method.

1. Analyze the truss using the method of sections or joint to determine the internal force N in each member. These forces are caused as a result of the application of the real loads only. Assume tensile forces as positive and compressive forces as negatives.

2. Place the unit load on the truss at the joint where the desired displacement is to be determined. The load should be in the same direction as the specified displacement, e.g., horizontal or vertical.

3. With the unit load so placed, and all the real loads removed from the truss, use the method of joints or the method of sections and calculate the internal n force in each truss member. Again assume the tensile forces as positive and the compressive forces as negatives.

4. Apply the equation of virtual work, to determine the desired displacement. It is important to retain the algebraic sign for each of the corresponding n and N forces when substituting these terms into the equation.

**Worked Example**

The simple truss shown is hinged at point A and on roller support at joint B. Determine the vertical and horizontal displacement of joint C. Assuming the cross-sectional area and modulus of elasticity of each member is constant.

The truss is first analyzed for the internal forces using the method of joints, and since the truss is statically determinate the equations of statics can be employed in determining these internal forces.

##### Support Reactions

Summation of all the horizontal forces in the structure must be equal to zero.

\sum { { F }_{ x } } =0

10-{ H }_{ A }=0----1

{ H }_{ A }=10kN\leftarrow

Summation of all the vertical forces in the structure must be equal to zero.

\sum { { F }_{ y } } =0

{ V }_{ A }+{ V }_{ B }=25kN----2

Summation of moment about point any point must be equal to zero. Therefore take moment about point A.

\sum { { M }_{ A } } =0

{ 16V }_{ B }-\left( 10\times 6 \right) -\left( 25\times 8 \right) =0

{ V }_{ B }=\frac { 260 }{ 16 } =16.25kN\uparrow

Substitute the vertical reaction at point B into equation 2

{ V }_{ A }=25-{ V }_{ B }=25-16.25=8.75kN\uparrow

##### Real Forces

Applying equations of statics to joint A and B we have

{ N }_{ AB }=21.64kN\quad ;\quad { N }_{ BC }=-27.08kN;\quad { N }_{ AC }=-14.58kN

##### Vertical Virtual-Force

To determine the vertical deflection of support C we apply a virtual unit load as shown in figure 4 and carry out the analysis for the member forces.

Applying the equations of equilibrium, we have :

{ n }_{ 1,ab }=0.667kN;\quad { n }_{ 1,bc }={ n }_{ 1,ac }=-0.833kN

##### Horizontal Virtual Force

To determine the horizontal deflection of support C we apply a virtual unit load as shown in figure 4 and carry out the analysis for the member forces.

Applying the equations of equilibrium, we have :

{ n }_{ 2,ab }=0.50kN;\quad { n }_{ 2,bc }=-0.625kN;\quad { n }_{ 2,ac }=0.625kN

##### Virtual-Force Equation

Since AE is constant, each of the terms nNL can be arranged in tabular form and computed. Here positive numbers indicate tensile forces and negative numbers indicate compressive forces.

Member | N (kN) | n1 (kN) | n2 (kN) | L (m) | n1NL | n2NL |

AB | 21.64 | 0.667 | 0.5 | 16 | 230.94 | 173.12 |

BC | -27.08 | -0.833 | -0.625 | 10 | 225.58 | 169.25 |

AC | -14.58 | -0.833 | 0.625 | 10 | 121.45 | -91.13 |

Sum | 578 | 251.24 |

Thus applying the virtual work equation for trusses, we have:

1.\Delta =\sum { \frac { nNL }{ AE } }

**Vertical deflection of point C :**

{ \Delta }_{ V }=\sum { \frac { { n }_{ 1 }NL }{ AE } = } \frac { 578 }{ AE }\downarrow

**Horizontal deflection of point C :**

{ \Delta }_{ H }=\sum { \frac { { n }_{ 2 }NL }{ AE } = } \frac { 251.24 }{ AE } \rightarrow

In this example, we used a simple truss to illustrate the principle of virtual work. However, its application encompasses beams, frames and even complex trusses with numerous members. Calculation by hand will no doubt be tedious however, a computer program can be written for the entire structure and then subsequently used to compute the displacements at the desired points. Most software packages of today are either based on this principle or an extension of this principle.

THANK YOU FOR READING! COMMENT & SHARE THIS POST

What’s up, after reeading this awesome post i am as

well delighted to share my experience here with friends.

Wow, amazing blog layout! How long have you been blogging for?

yoou made blogging look easy. The overall look of your web site is excellent,as well as the content.

Hello, I enjoy readijng all off your article.

I like to write a little comment to support you.

With thanks for sharing your superb website!

Hey! This is my first comment here so I just wanted to give a quick shout

out and say I truly enjoy reading through your articles.

Apprreciate it!

Ohh, its fastidious discussion about this article here at this web site,

I have read all that, so noww me also commenting at this place.

Hi there to every body, it’s my first pay a quick visit of this website;

this blog contains awesome and genuinely excellent data ffor visitors.

Really interesting information, I am sure thus

post has touched all internbet users, its really really pleasant piece

of writing on building up new website.

Please keep us up tto date like this. Thanks for sharing…

With thanks for sharing your superb website!

Very nice post. I just stumbled upopn yoour blog and wanted to say that

I’ve trhly enjoyed surfing around your blog posts. In any

case I will be subscribing tto your feed and I hope you write

again very soon!

Well composed articles like yours renews my faith in today’s writers.You’ve written information I can finally agree on and also use.Many

thanks for sharing.

I enjly what you guys are usually up too. This tyoe of clever work and reporting!

Keep up the very good works guys I’ve added you

guys to my own video.

QS

It’s not my first time to go too see this web page, i am visiting this web site very often and ake good factrs from here.

Hello There. I found your blog using google. This is ann extremely wwll written article.

I’ll make sure too bookmark it and return to read more of

your useful information. Thanks for the post. I’ll certainly return.

I’m very happy to discover this page. I need

tto to thank you ffor ones time just forr tyis fantastic read!

I definitely really liked every little bitt of it and I have

yoou bookmarked to check out new things in your blog.

Your mode of describing all in this piece of wriing is in fact pleasant,

all be capable of simply be aware of it, Thanks a lot.

I got this site from mmy pal who shared with me concerning this website

and now this time I am browsing this website and reading very informative articles or reviews at thgis

place.

I do not even kniw how I ended up here, but I thought this post was great.

I don’t know who you are but certainly you are goinmg tto

a famous blogger if you are not already 🙂 Cheers!

Hello! Do you use Twitter? I’d like to follow yyou

if that would bee okay. I’m definitely enjoying your blog and looik forward too

new articles.

Hello tto every one, sincce I am genuineely keen of

reading this website’s post to bbe updated regularly.

It carries nice data.

Great delivery. Great arguments. Keep up the amazing spirit.

Wow cuz this is great work! Congrats and kesp it

up!

Please keep us up to date like this. Thanks for sharing…

Hi there to every body, it’s mmy firs pay a quick visit of this website; this blog contains awesome and genuinely excellent datra

for visitors.

It’s nott my first time to go to see this web page, i am

visiting this web site very often and take good facts from here.

I am pleased that I detected this web blog, just the right information that I was searching for!

Very good post! We will bbe linking to this great content on oour website.Keep up the great writing!

Fabulous, what a web site it is! This web site provides useful data

to us, kdep it up.

What’s up, I wish for to subscribe for this blog to take most recent

updates, so where can i do it please help out.

Hello all, here every person is sharing such know-how,

soo it’s pleasant to read this website, and I used to go to see ths website daily.

Your mode oof describing all in this piece

of writing is in fact pleasant, all be capable of simply be aware of it,

Thankks a lot.

I enjoy what you guys are usually up too. This type of clever work and reporting!

Keep uup the verry good works guys I’ve dded you guys to my

own video.

Pretty! This has been an extremely wonderful post.

Thanks for providing tis info.

I like it whenever people come together and share thoughts.Great website,

continue the good work!

Very nice post. I just stumbled upon your blog and wanted to say that I’ve trulky enjoyed surfing around your blog

posts. In any case I will bbe subscribing to your feed and

I hope you write again very soon!

Hello, after reading thos amazing article i aam as well happy to share

my familiarity hede with mates!

Very good post! We will be linking to this great content on our website.Keep up the great writing!

Hi tere to every body, it’s my first pay a quick visit of this website; thhis

blog contains aweesome and genuinely excellent data for visitors.

cialis online fast shipping – cialis online canada best online thai pharmacy

us online pharmacy generic viagra – best female viagra over the counter how much is a viagra prescription

tadalafil pills price – 20 mg cialis cost cvs how to buy cialis over the counter

ivermectin 1mg – stromectol 12mg online can you buy stromectol over the counter

casino world – real money casino online online casinos

fda approved over the counter ed pills – erectile dysfunction medicine otc ed pills that work

prednisone in mexico – order prednizone how to get prednisone without a prescription

cialis 10mg tablet – Cialis on line purchase where to buy cialis online canada

stromectol 3mg online – cost of stromectol lotion stromectol 12mg

Дом Гуччи смотреть фильм онлайн

generic ed pills – ed vacuum pump erectile dysfunction pump

в хорошем качестве смотреть фильмы онлайн бесплатно

ventolin 90 – ventolin uk price ventolin pills

where to get cytotec pills – cytotec 200mcg tab where to buy cytotec over the counter

doxycycline mexico – doxycycline 30 doxycycline hyclate 100 mg

neurontin 300 mg tablets – synthroid 0.112 mcg synthroid 100 pill

pfizer viagra – cheap generic viagra india purchase cheap viagra online

http://buysildenshop.com/ – Viagra

can you buy cialis otc in canada – buy online pharmacy uk generic cialis no rx

De Amoxil Amoxil Po De Suite

generic vardenafil – buy vardenafil line vardenafil

Whoesale Levitra Pills

proscar no script

buy cialis online prescription

stromectol tab 3mg – buy liquid ivermectin where to buy ivermectin pills

compare viagra cialis

https://buystromectolon.com/ – Stromectol

Stromectol

Levitra Original Pas Cher

where can i buy prednisone without prescription – prednisone 20 mg brand name prednisone online purchase

accutane without a prescription – accutane 10mg daily accutane 20mg

смотреть

amoxicillin brand name – antibiotics without a doctor’s prescription amoxicillin brand name

methylprednisolone 80 – medrol cost lyrica south africa

viagra 20 mg – Buy pfizer viagra buy viagra

Viagra Valor

cialis 20 mg price in canada – How to get cialis cialis 30

Клятва врача

Злое

Дом на другой стороне

Воспоминание

ДЮНА

Соври мне правду

Кэндимен

4408

3193

9344

http://buytadalafshop.com/ – where to buy cialis online safely

stromectol cvs – stromectol cost australia ivermectin uk coronavirus

order prednisone 40mg online – 50 mg prednisone online prednisone 2.5 mg cost

85

furosemide 20mg uk – how much does clomid cost lasix without prescription usa