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Structural Analysis III

Lecture Notes

  1. Course Introduction
  2. Basis for the Analysis of Indeterminate Structures
  3. Characteristics of Structures
  4. Qualitative Analysis
  5. The Moment-Area Method
  6. Moment Distribution
  7. Macaulay’s Method
  8. Virtual Work
  9. Plastic Analysis

Lecture Notes for 2010/11

Semester 2

  • Macaulay’s Method for Indeterminate Structures: The deflections of indeterminate flexural structures. Updates of last year’s notes.
  • Virtual Work: updates and some shiny new examples and explanations.
  • Plastic Analysis – a few extra bits and one piece.

Semester 1

  • Course Introduction notes;
  • Updated Qualitative Analysis notes for the analysis of structures without numerical calculation.
  • Basis for the Analysis of Indeterminate Structures here.
  • Updated Moment Area – Mohr’s Theorems Notes here.
  • Updated Moment Distribution notes here.

Lecture Notes for 2009/10

Semester 2

  • Macaulay’s Method for Indeterminate Structures: The deflections of indeterminate flexural structures. Minor updates of last year’s notes.
  • Virtual Work…again only minor updates unfortunately.
  • Plastic Analysis – new examples and extra explanation make this a thrilling read!

Semester 1

The Lab notes for, well, the laboratory work.

Qualitative Analysis notes for the analysis of structures without numerical calculation.

Basis for the Analysis of Indeterminate Structures – updated from last year’s set – background concepts and commonly used assumptions are explained;

The Moment Area Method notes: lots of updates – hopefully much more interesting! Supporting downloads are:

Updated Moment Distribution notes.

Lecture Notes for 2008/9

Semester 2

  • Macaulay’s Method for Indeterminate Structures: The deflections of indeterminate flexural structures. An updated set of last year’s notes.
  • Virtual Work: Used to analyse frames and trusses for moments and deflections. An updated set of last year’s notes.
  • Plastic Analysis: The calculation of the actual collapse load of structural frames, given the capacity of the members. An updated set of last year’s notes.
  • Semester 2 Exam information for the revised format;
  • Plastic Bending of Beams Lab workings;
  • UPDATED set of Plastic Analysis notes, with an extra example.

Christmas Spaghetti Bridge Competition Results 2008/9

Group Bridge Weight (kg) Strength (kg) SW Ratio Prediction (kg) PS Ratio
1 0.861 4.81 5.59 7.0 1.46
2 0.820 9.01 10.99 8.0 0.89
3 0.859 4.24 4.94 15.0 3.54
4 0.641 6.48 10.11 10.0 1.54
5* 0.327 0.97 2.97 3.0 3.09
6 0.857 1.9 2.22 2.3 1.21
7* 0.971 8.14 8.38 10.0 1.23
8* 0.523 1.07 2.05 1.5 1.40
9* 0.400 1.5 3.75 2.0 1.33
10 0.787 6.88 8.74 10.0 1.45
11 0.885 7.86 8.88 3.5 0.45

* These groups were disqualified due to non-conformance with the specifications of the problem.

First, second, and third were awarded to Groups 2, 4 and 11 respectively (as may be seen by sorting on the SW column). The Faculty Director also awarded Group 5 a special merit prize for their unique entry.

One very intersting aspect of the results is the significant over-estimation of strength across the board, save for Groups 2 and 11. Is it coincidence that these groups were prize-winners?

Semester 1

Lecture Notes for 2007/8

  • Mohr’s Theorems: Used to calculate deflections and reactions of determinate and indeterminate structures.
  • Moment Distribution: Hand method for the analysis of indeterminate structures.
  • Structure Characteristics: The identification of the type of structure being analysed and whether it is stable or not and the movement that structures may undergo, covers Stability; Statical Determinacy and Indeterminacy, and; Kinematic Indeterminacy.
  • Virtual Work: Used to analyse frames and trusses for moments and deflections.
  • Qualitative Analysis: Estimating bending moments etc. without doing any calculations.
  • Macaulay’s Method for Indeterminate Structures: The deflections of indeterminate flexural structures.
  • Plastic Analysis: The calculation of the actual collapse load of structural frames, given the capacity of the members.
  • Compatibility and Superposition: some of the very basics upon which the analysis of indeterminate structures is based.


  • 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.
  • Macaulay’s Method for a beam with a hinge – the MS Excel file from the example in the notes.
  • Virtual Work for continuous beams – the MS Excel file from the example in the notes.
  • Total Potential Energy of a simple spring system – graphs from the Virtual Work notes.


Comment from Walid
Time: 29 May, 2009, 23:06


You have excellent course notes.

On the subject of yield analysis, there are many examples here and elsewhere regarding rectangular slabs with different boundary conditions, i.e., end fixity. When it comes to analyzing a triangular concrete slab, how can one set up the external work equal to the internal work? How to compute the maximym out of plane deflection of the top of the barrier. Do you have an example problem? If so, please post.

Your assistance is gretly apprecited!!!

Note: I was referring to a concrete bridge barrier subject to a transverse impact load. Based on the yield analysis suggested by Hirsch, th failure will be in the form of a triangular slab. The area of failure will have moments along the yiled lines, thus plastic hinges develop. The side of the wedge may be considered as fixed while the top of the barrier (outer side) is not fixed.

Comment from Ryan
Time: 19 June, 2009, 17:24

This is a great site. Thanks for all your posts. I like this site so much. =)

Comment from Barbra
Time: 21 July, 2009, 02:24


I was just researching for my assignment on Structural dynamics and your lecture notes are most appropriate. I am so impressed with your order and quality of information and detail in your notes. Your efforts are not in vain, I personally appreciate you sharing your knowledge so freely. Ryan is right, this site is Great! I am going to be visiting it quite frequently. Thanks and God bless!

Comment from Marco
Time: 14 October, 2009, 21:52

Your notes are unbelievably helpful!
Thank you so much, I might just pass structures after all!

Comment from Suganya Paskaran
Time: 3 January, 2010, 11:53

Hello Dr. Caprani,
I am a level 3 Civil Engineering student in sri lanka.
Your lecture notes are very useful to my course.
thanks a lot for the great service.
thanks for the the free access.

Comment from sham
Time: 12 February, 2010, 09:05

I m studying civil engineering in malaysia and your notes really helped me on my test.
many thanks indeed.

Comment from Hugh
Time: 23 February, 2010, 12:19

Hello Dr. Caprani,

I have already completed my Msc and find the notes extreamly helpfull!! Thanks for posting them. A good read!

Comment from Jim
Time: 20 April, 2010, 18:34

Wow Dr.Caprani, these notes are all great, I have got my 3rd year Analysis exam soon, I’m glad I found them. Please come back to UCD!

Comment from Nkanyezi Mazibuko
Time: 5 May, 2010, 12:33


I would like to inquire about the stiffness matrix method.How does one deal with combined structures with regards to stiffnesses of different elements and inputting them into the global stiffness matrix. e.g a beam with spring supports

Comment from Admin
Time: 5 May, 2010, 20:18

Refer to my 4th year notes here. For a beam with spring supports the spring stiffness is added to the main diagonal corresponding DOF. See how this is implemented in the program Continuous Beam Analysis here. The references in my notes should also be of assistance.


Comment from Mike
Time: 25 June, 2010, 14:55


Do you have any notes on taperd reinforced concrete columns? If not, do you know where i could find these?

Comment from Dratre Anthony
Time: 31 August, 2010, 14:51

I have found your notes helpful for my studies, kindly forward for me your notes on structural anlysis III and IV

Comment from Parveshen Naidoo
Time: 6 March, 2011, 12:59

thank you for the notes Dr Caprani, i used the qualitative set quiet thoroughly. we really appreciate your help in our structural analysisIII course

Comment from Steve
Time: 25 April, 2011, 10:31

Superb site, it is very rare to find such an open site offering so much quality information. Thank you

Comment from Jyoti Prakash Mahanta
Time: 6 May, 2011, 13:51

I’m studying Civil engineering in India. Your notes are awesome, hope you will include Analysis of various Bridges in your course.

Comment from Admin
Time: 6 May, 2011, 15:22

Thanks Jyoti,
Hopefully! There may be a Masters course soon that I’ll be able to do some work on bridges in. Thanks for the nice comment.

Comment from Nkanyezi Mazibuko
Time: 19 May, 2011, 07:44

Hi i would like find inquuire about supports in the moment distribution. If you have a slider support at one end of a beam that is inclined, do you 1st treat it as a fixed support in order to obtain the FEMs, if so how do you account for the vertical reaction as a result of fixing the support?

Comment from Arun kumar
Time: 27 June, 2011, 06:33

a very gud site . . but i need some idea to prepare best poster presentation pls help me

Comment from hadi bahmani
Time: 29 November, 2011, 19:56

I am so pleased to find your website.Your material is very useful.Thank you.

Comment from asimsos
Time: 13 December, 2011, 19:32

hi dr capriani. i am studyin civil engineering in TURKEY and i need your help. in this term i am having structural analysis project lesson. my homework is “solving 7. degree indeterminate system with stiffness matrix method” . first i need summarize methot but not academicly (studently) 🙂 i mean it should be apllicable maybe include solved some questions…. PLEASE HELPP MEEE. E-MAIL :

Comment from Amad Afzal
Time: 20 January, 2012, 15:49

Dr. Caprani.

I just want to say a big thank you to you on behalf of me and my course mates. Your notes are of the highest standard and I feel that every university in the world should use these to teach structures 3 & 4. If I ever get the honour of meeting you I would like to give you a big hug as these notes are the only reason why I will ever pass my structures module.

Thank you and Many Regards.

Comment from Admin
Time: 20 January, 2012, 18:49

Thank you so much for your kind comments – they are very encouraging 🙂 I hope you do well int eh rest of your course and go on to have a great career in engineering – there really is nothing like it!

Comment from Admin
Time: 20 January, 2012, 18:51

Have a look at the Structural Analysis 4 notes. Honestly, I’m not holding back on some “magic sentence” that will help students understand the material – whatever I can offer is in the notes. Reading notes from different authors on the same topic has always helped me, as well as hours and hours practicing problems.
Good luck,

Comment from Ethan
Time: 11 April, 2012, 13:36

Dr. Caprani,

This website is great! Full of usefeul information. I just have few questions about your 2011/12 Virtual Work note. At the end of page 30, there is a equation states that delta y = delta e1 = 3/5*delta e2. Should it be delta y = 5/3*delta e2 instead? Otherwise, the next equation doesn’t make sense to me. Similar thing happens to the equation at the end of page 31. Should it be delta y = 5/4*delta e2 instead?

Comment from Admin
Time: 11 April, 2012, 20:02

Very nice spot! Yes, the equation on the bottom of page 30 should read delta y = delta e1 = 5/3*delta e2 since the delta y is the hypotenuse of the triangle and the delta e2 is the “3” side of the 3-4-5 triangle. The next equation is then correct. The same issue as to the orientation of the displacement triangle resolves the 4/5 problem too.

Thanks for letting me know – keep them coming if you spot any more!

Comment from Kuttan
Time: 30 April, 2012, 04:14

Excellent notes

Comment from T.Rangarajan
Time: 14 May, 2012, 01:55

Dear Dr.Caprani,
It is grate work and a service to all structural engineers. Even though I am 67 still I learned a lot your notes.
I as well as other structural engineers who would like to learn in depth owe debt of thanks.

Comment from suchit
Time: 18 June, 2012, 09:57

Dear Dr.Caprani,
It is a very great thing that sharing one’s knowledge to others. That to with practical way and to the great extent. I found this web site is very useful for all structural engineers. It is really a great service and kindly accept my sincere gratitude for the valuable work and time.

Comment from Kofi Kwarteng Opoku
Time: 19 June, 2012, 20:16

please, could you mail me on how to design an electric mast?

Comment from mark
Time: 10 August, 2012, 10:16

HI Mr. Caprani

On your lectures notes for the Moment Distribution diagrams 2008/2009, specifically problem 8( titled 2006) on non-sway frames, the force on VA on lectures is 40.3. Can you tell me why this is? i have done my calculations but i got 94 for my VA.

Thank you


Comment from Admin
Time: 13 August, 2012, 13:17

Nice spot – yes that was an error. All other reactions are correct. Vd and Va should sum to 40 x 4 = 160 kN.

Comment from Chris
Time: 7 September, 2012, 21:21

hi Dr. Caprani
i am a final yeah civil student in South Africa. i am struggling with a problem and have exhausted every possible method on how to solve it.
it deals with Moment Distribution.
how to incorporate a slider as on of the end supports, is this a sway structure or not? and how do you do it..haha
And thx for you note.
Matrix Methods was impossible untill i saw yours!

Comment from Chris
Time: 7 September, 2012, 21:22

if anyone else knows how to do this kind of problem help would be much appreciated!!

Comment from structures01
Time: 1 October, 2012, 09:50

Are there any memos available for the past question papers in the notes?
Thanks for the great work

Comment from anon
Time: 26 November, 2012, 07:14

thank you so much for your notes. i had forgotten so much and they were an effective refresher. please post more

Comment from Anonymous
Time: 9 December, 2012, 17:27

Cannot believe you have made your notes so freely available, thanks very much, these notes are excellent, its a pity you could not come down to IT Carlow and give us some time and help because we could do with it. Well done again, you don’t realise how helpful this is.

Comment from george
Time: 19 December, 2012, 13:07

Hello thanks alot for your notes on plastic analysis.Do you have solution to the pass exam question.Thanks George from Kenya.

Comment from mohammad srafili
Time: 21 December, 2012, 21:14

plastic moment of sections

Comment from Tony Reeve
Time: 15 January, 2013, 23:42

Is there any chance of getting a solution for 3 rd year structural engineering 2010/11 setion 3.4 problems 4 & 5 (inderminate beams)? Have used Macaulay’s for years to solve deflection problems for single span beams and shafts, now have a problem to determine reactions on 2 span beam with irregular loads & I wanrt to use Macaulay’s method to solve.

Comment from chukwudi prince
Time: 24 February, 2013, 02:05

thank you very very much dr. colin caprani, please i need a material on slope deflection method

Comment from Moses Mashinini
Time: 24 April, 2013, 16:00

Hi Mr. Caprani. I must say your notes have been very helpful in ways i never imagined. I am now doing my final year and have a few questions I need you assistance with.
1. How to deal with springs in Plastic Analysis i.e. given a spring support and its moment capacity.
2. How to find the carry over factor in moment distribution for a spring end and a slider end.

Thank you in advance
Moses Mashinini

Comment from Rodolfo Rodriguez
Time: 4 July, 2013, 17:10

Dear Professor , does professor have any solved examples of moment area method taking the moment area by cantilever parts?
Best regards ,
Rodolfo Rodríguez

Comment from Admin
Time: 5 July, 2013, 09:38

Hi Rodolfo,
See Example 11 of the Moment Area lecture notes here (no. 5).

Comment from Admin
Time: 5 July, 2013, 09:44

These are good questions and are more advanced applications of the theories – and in actual fact allow more realistic modelling of structures. To answer partially:
1. I think an incremental analysis is required in this case. A spring can theoretically still allow the plastic moment capacity of the connecting member to be developed if it is sufficiently stiff such that the rotations are small enough. However, if it is not stiff then very large rotations would need to be developed. These could only occur if other parts of the structure had already yielded. Therefore an incremental approach is probably required. I don’t have any reference in my mind to assist you with this, though Prof. Bill Wong’s book is very good on Plastic Analysis.
2. See Gere’s book on Moment Distribution (referenced in my notes). You could actually work out the real carry-over-factor using any approach (e.g. virtual work, moment area). Then proceed with the moment distribution table as normal, but using the altered carry over factor for the member with the spring at the end. As for the slider, well if rotation and vertical movement are suppressed then it makes no difference.

Comment from yihimalle
Time: 14 June, 2015, 11:07

how can i get the solution to the questions on the lecture note

Comment from V.Anh
Time: 29 June, 2015, 04:49

Its such a great site with amazing read. I first knew your page since I was in Structural dynamics course last year, now I come back for my final project. And yes, I agree with Amad Afzal that your notes are the highest standard. To me, those notes are truly a rich source of knowledge, skills and mostly, the inspiration, I can feel the sheer joy of being a learner.
Thanks for all.

Comment from BlessKotazz
Time: 18 September, 2015, 10:26

Thank you for such well structured and beautiful notes. You’re Structural Analysis notes really helped me!!! Thanx Doc

Comment from Ngoni
Time: 8 August, 2017, 07:44

Hi Dr Caprana,

I always enjoy using your notes .

I have one problem on Direct stiffness where I have a frame with both joints pinned meaning that there is no Final moment on my reaction, How do I eliminate this or else I should just follow the procedure and the Z values will automatically zero themselves.

Comment from Shams Ul Islam
Time: 26 July, 2018, 19:39

Hi Dear Dr. C. Caprani.
I really appreciate your effort Sir and Thanks for such beautiful notes. Sir, i have some problem in the solution of “Plastic analysis 3rd year Structural Engineering 2009/2010 notes page no.109 (Past exam Paper Q.No. 7.1 Summer 2000). Kindly send me the solution of the said question.
Shams Ul Islam
Lecturer, CIvil Engineering Deptt. CECOS University Peshawar Pakistan

Comment from Mukuka
Time: 13 August, 2018, 00:12

Hi Dr Capran,
The notes are great.Thanks

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