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Structural Analysis IV


Matrix Stiffness Method R2.

Structural Dynamics R2.

Lecture Notes for 2011/12

  1. Course Introduction
  2. Virtual Work: Compound Structures
  3. Virtual Work: Advanced Examples (will not be gone through in class)
  4. Matrix Stiffness Method
  5. Structural Dynamics

Lecture Notes for 2010/11

Compound Structures using Virtual Work

  • Notes here – quite a lot of updates.
  • A proof of the means of using the integration tables for part UDLs here.

Matrix Stiffness Method

  • Updated notes here.
  • Matlab scripts for Analyze Truss here.

Lecture Notes for 2009/10

Matrix Stiffness Method

Nice pretty new notes here.

A beam analysis spreadsheet based on the stiffness method.

Structural Dynamics

Solutions to Examples – please try the example problems though before you view the solution.

Extra: explanation of the amplitude solution to the equation of motion.

Major overhaul of the Structural Dynamics notes for 2009/10 here.

Software to complement the notes for download:

Combined Structures

Moderately updated from last year:

  • Here are the computerized notes;
  • Here are some miscellaneous examples.

Lecture Notes for 2008/9

Computer Modelling of Structures – Group Projects – 2008/9

The projects are available as follows:

  • Group 1 – Not yet available;
  • Group 2 – Nonlinear analysis of an RC beam;
  • Group 3 – Shear walls with openings;
  • Group 4 – Arching action in curved masonry walls;
  • Group 5 – Flat slab design using nonlinear FE and traditional methods;
  • Group 6 – Investigation of strip footings under column loads;
  • Group 7 – Convergence of FE to theoretical result for plane stresses in a beam;
  • Group 8 – Not yet available;
  • Group 9 – Calculation of beam deflections subsequent to a stiffness analysis;
  • Group 10 – Vibration absorption for an SDOF system;
  • Group 11 – Not yet available.

Semester 1 Exam Handout.

Lecture Notes for 2007/8

These notes were developed based on Robert Mahony’s notes delivered to DIT for over 20 years. They are not computerized but scanned in acetates but they may still prove useful.

  • Combined Structures: Virtual work analysis used for structures whose members undergo bending and/or axial force.
  • Analysis of Arches: Virtual Work for moments/shears and axial forces in parabolic and semi-circular arches.
  • Stiffness Method: The basis of modern structural analysis software. A particular case of FE.
  • Influence Line Analysis: Used to determine design loads for members in structures subjected to moving loads (e.g. bridges) or for repeated analysis of a structure under various loading scenarios.
  • Structural Dynamics: Exact and approximate methods of determining the motion of structures under dynamic loading scenarios.


LinPro is great for analysing 2D structures. There are no bells & whistles to distract from the basic analysis and you can see the stiffness matrix for each member. The interface is quite easy (apart from the load definitions which are strange!). Well recommended for checking hand calculations and vice-versa.


Comment from Neil Kempton
Time: 21 November, 2008, 12:46

Great notes on structural dynamics. At the risk of sounding like a smartarse, have a look at p34. solution to cos(x)cosh(x) + 1 = 0 will be n(pi) except that n will be a non integer determined numerically? This is unlike the s-s example.

Comment from Colin
Time: 21 November, 2008, 13:53

That’s right – aL is a non-integer – as found from the roots of eq 4.28. Knowing L, substitute aL into eq. 4.21 to get w = a^2*(EI/m)^0.5. The first three roots, or values of aL, are given in Fig. 4.4.

Comment from sintayehu
Time: 11 March, 2009, 14:08

I loved your site for its completeness. I’m now studying civil engineering in Ethiopia(Jiimma University). It is my request if you can possibly send me materials for we have almost none in the University.

Comment from Colin
Time: 11 March, 2009, 14:16


Thanks for your kind comments. Feel free to download anything of interest from the site.

Comment from Ali El-baden
Time: 2 January, 2010, 22:40

it is one of the best web site i have seen

Comment from Liz
Time: 23 January, 2010, 20:35

Best site ever! I don’t know what I’d have done if I hadn’t found this site. Your notes are amazing!

Time: 3 July, 2010, 11:59

From now I got the full knowledge to help me in my study
I thank you a lot for this hard work that you have done.

Comment from Constantinos Kalotaris
Time: 19 August, 2010, 18:27

For those interested in structural dynamics also the notes of Prof Lynch from University of Michingan Ann Arbor

Comment from umesh p patil
Time: 27 September, 2010, 12:24

dear dr colin ..i am working as assitant professor in the prestigious KLE engineering college belgaum,INDIA U have done a commendable service by providing these material online…I have gone through lecture note and liked the way u presented them..thanx a million

Comment from Colin
Time: 22 October, 2010, 20:56

Thanks for your kind comments!

Comment from Norman
Time: 30 October, 2010, 03:21

Dear Colin, without your notes I could have never managed second year- BEng-civil, the way you present your concepts, it just makes things easy. Thanks for your generosity.

Comment from Steve O’Callaghan BEng (Hons) M.ASCE MCIArb CEng MIEI
Time: 3 March, 2011, 20:13

This site is truely a credit to you. If only there were more engineers like you who like to see our profession progress the way you do. You are an inspiration.

Comment from RASAQ KEHINDE
Time: 8 March, 2011, 11:54

Dear Collin. it,s a great pool of knowledge in structural analysis. it had really improve my knowledge of Analyzing indeterminate structures. Thanks

Comment from Arun
Time: 15 April, 2011, 07:46

Dear Sir, It’s great to see such an open minded person like you. your web page is really wonderful.

Research Scholar, India

Comment from kk
Time: 17 June, 2011, 03:34

can i get spreadsheet for analysis of coninuous beam using matrix stiffnes method……..

Comment from Jay
Time: 7 July, 2011, 06:33

hii all , I m student.
If anyone have visual basic program and excel sheets for design and punching shear check of flat slab please send it on my e-mail.

Comment from Jay
Time: 7 July, 2011, 06:34

My e-mail id is

Comment from John
Time: 20 October, 2011, 14:56

I’d like to thank you for sharing all this information online. The notes you’ve posted online especially for Structural Analysis III have greatly assisted me.

Comment from Alex
Time: 30 November, 2011, 08:53

Each and every single note that you have posted was great. I hope, so many students all over the world would enjoy it. Thank you so much.

Comment from JOHN85
Time: 8 December, 2011, 23:51

I would like to thank you so much for sharing all this stuff and notes, especially on Structural Analysis. However, i would be grateful if you could assist me in the following: elastoplastic (non linear) SDOF system analysis in MS Excel, using Newmark’s linear acceleration numerical method. It could be extremely useful if you d post or mail to me a procedure in Excel for A.K.Chopra’s Dynamics of Structures, Example exercise 5.5, page 190. I cant find out how can i check whether the system is behaving elastically or plastically for each time step, so that we use ki=k or ki=0 and fsi=…… or fsi=fy respectively.

Comment from Tisna Sanjaya
Time: 11 December, 2011, 17:31

Nice, thanx a lot for ur knowledge, hope in future u can more succes,

Comment from cherry
Time: 3 January, 2012, 22:46

hi thanks for sharing notes,… can you share to me some notes in buckling, elasticity and incremental plastic analysis?

I am taking masters in civil engineering at universita di bologna italy.

cherry moreno

Comment from Eder Fuabuna
Time: 9 March, 2012, 10:26

Dear Doctor Caprani,

I am Eder Fuabuna,i am studying engineering civil(structural engineering=b-tech civil) in south africa,it is my first time to study in english institution and i have completed my diploma in the democratic republic of the congo.I would like to get the not of the analyse of the slab,method of the yield line and also the plastic analysis and the matrix analysis notes 3 and 4,can i get registered to get the notes,where?

Comment from Admin
Time: 9 March, 2012, 12:48

All of the notes you want are available to download from the pages on this site – just have a look around. You don’t need to register.

Comment from Dan
Time: 17 May, 2012, 02:33

Fantastic resources here. Wish my lecturers were producing excellent notes like you do, Dr Caprani.
BTW, I downloaded the matlap script for the Stiffness Method. After I enter the data for nData and eData, and run the script as instructed in your notes, I get two Matlab errors:

??? Error: File: AssembleTrussK.m Line: 6 Column: 6
Unbalanced or unexpected parenthesis or bracket.

Error in ==> AnalyzeTruss at 6
kg = AssembleTrussK(nData, eData); % Assemble global stiffness matrix

It seems that in the first error Matlab considers the “~]” in [ne ~] to be non Matlab recongizable syntax. How do I fix this? I am running Matlab 2007 R14.

Hope you can help.

Comment from Admin
Time: 17 May, 2012, 15:25

Thanks for the kind comments – much appreciated, and I’m glad you find the material useful. The tilde operator was introduced in R2009b as a way of avoiding creating dummy variables that are not required in a script. A good explanation is here. To fix the error above write: [ne dummy] instead of [ne ~]. The dummy variable just isn’t used anywhere in the script.
Hope that helps,

Comment from salar
Time: 3 June, 2012, 12:22

hi,i’d be glad to have your email sir,to sending you some Excel program which i wrote can be useful for civil student

Time: 10 June, 2012, 05:55

Your available notes on structural analysis helped me passed my third year course. May you be richly blessed.A Ghanaian University Student.

Comment from Bassam Daoud
Time: 2 July, 2012, 18:03

Dear Dr. Caprani,
I really thank pioneers such as yourself to fill in the gaps in practicing engineers’ knowledge and I must say that you markedly increased my knowledge with respect to structural dynamics. I wish I was a student of yours as dynamics with respect to structures is poorly taught elsewhere and with little emphasis. And the sad thing is that companies frequently ask potential job seekers if they knew how to analyze a structure dynamically which leaves a lot of engineers stumped. And the companies are quite right as they do not want a structure to resonate or have excessive vibration when designed lacking a dynamics check. Thank you for furthering engineering.

Comment from Admin
Time: 2 July, 2012, 21:30

Thank you for your very kind comments. There are some other great sites for structural dynamics also. I’ll put some in the links section soon.

Comment from klark
Time: 3 August, 2012, 09:49

best website! Thanks!

Comment from aref from iran
Time: 28 August, 2012, 22:10

thank you for every thing that you shared it with all student in the world,those maybe couldnt acces this in their country or university.i hope to be yor student.

Comment from sarwar
Time: 3 October, 2012, 14:59

Dear Doctor Caprani
it is one of the best web site i have seen,I’d like to thank you for sharing all this information online. The notes you’ve posted online especially for Structural Analysis III have greatly assisted me,but now I can not download the lectures for forth years How can I do this .

Comment from wilson
Time: 21 December, 2012, 04:11

Great man with great knowledge thanks for sharing all that you known with the rest of the world

Comment from wilson
Time: 23 December, 2012, 03:29

Thank you sir for sharing your knowledge for free.

Comment from DEVI PRASAD
Time: 19 March, 2013, 09:54

Hats off to you dear Dr.Colin….

Comment from Sameh Mahfouz
Time: 23 March, 2013, 18:12

Dear Prof. Colin
I do thank you for the helpful website. All in all, you are a good man.
Prof. Sameh Mahfouz

Comment from Abdul
Time: 2 September, 2013, 12:14

Dear Dr Caprani
The notes posted here on arches have proven very useful for me. I would like to request from you any links to other possible sources on arches which deal with arches such as two pinned bows and temperature effects etc. My efforts thus far have yielded very little.

Comment from Abdul
Time: 2 September, 2013, 14:29

Dear Dr Caprani
I have found your notes on arches very useful. I would much appreciate it if you could provide me with an indication of other source/reference material which covers indeterminate bow arches and temperature effects on arches etc. I have tried to locate such material but with very little success. Any help would be much appreciated.

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