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Beam Theory

What is the advantage of using Timoshenko beam theory over other beam theories? Also what is the
differnce between a Beam, a thick plate and a thin plate. Iam new to
this area so these questions may sound stupid to you. Also i want to know
that if a plate is thick then what will be the solution approach and same
for a Beam and thin plate. lets say we have to find deflection in all cases
(Beam, thick and thin plate)Thanx Sandyg

zishun liu's picture

Dear Sandyg

Timoshenko's theory of beams constitutes an improvement over the Euler-Bernoulli theory. The difference between the Timoshenko beam and the Bernoulli beam is that the former includes the effect of the shear stresses on the deformation. A constant shear over the beam height is assumed. It is also said that the Timoshenko’s beam theory is an extension of the Euler-Bernoulli beam theory to allow for the effect of transverse shear deformation. Timoshenko’s beam theory relaxes the normality assumption of plane sections that remain plane and normal to the deformed centerline. For example, in dynamic case, Timoshenko's theory incorporates shear and rotational inertia effects and it will be more accurate for not very slender beam.

For your second question, what is the difference between a beam, a thick plate and a thin plate; it depends on the relative dimensions of length, width and thickness of the structure. When is a beam or plate of a structure? It is depending on the ratio of beam-width-to-beam-depth. For more details, you may refer to any text books, such as “Strength of Material” or “Plate and Shell Structures”.

Dear Dr. Zishun,

I have read a few of your  interesting papers on finite element analysis, focussing on the nine noded assumed strain shell element for dealing with contact impact problems. I am a research student of NUS working on higher order finite elements. I have developed a higher order triangular plate element with 135 d.o.f per element and tested the element for cases with lesser boundary constraints. The element is found to predict stress resultants extremely well (satisfying all natural boundary conditions and showing no oscillations in stress distribution - even for very thin plates). I have developed the element using MATLAB.

Due to memory constraints and difficulties in handling complex shapes, I would like to include the finite element formulation of this new element in ABAQUS.

Kindly let me know if this would be possible to implement in ABAQUS. If so, kindly give me an outline of the procedure so that I start working.

Thanking you in advance!!!

Sai Sudha Ramesh

zishun liu's picture

Dear Ramesh,

If MATLAB is not suitable for your problems, I suggest that you may use COMSOL software to solve it. This is more flexible for solving your problem. As you are familiar with MATLAB, it should be no any problem for you to use COMSOL. Of course you may try to use ABAQUS to solve it, in my opinion, it may have more credit for your research.



ramdas chennamsetti's picture

R. Chennamsetti, Scientist, India

In continuation of Dr. Zishun Liu's, we incorporate shear correction factor, to take care of the shear strain energy, which is supposed to be calculated by considering actual shear stress distribution across the beam depth.

You may also refer "Theory of Elastic Plates" - J. N. Reddy

zishun liu's picture

That is right. You may refer my colleagues Prof. CM Wang and Prof. JN Reddy's paper on this problem. I am sorry I can't remember the details, but you can find it from Prof. CM Wang's publications.

Hi Sandyg,
At first, S.P.Timoshenko and G.M.Gere - world super stars of Applied Mechanics.

There is one little difference - shear deformations that we added to Euler bending. More particulary we understand that we have a bending angle and deflection as deformations at beam bending process. But accuracy requires to add a shear angle to bending angle. Then we have a little more total angle than bending angle only.

 This example is reproduced in the school laboratory of one of technical Universities of Belarus. The experiment confirms brilliantly the improved theory of S.P.Timoshenko, taking into consideration the shear deformations. The parameters of cross-section area correspond to the flange beam ¹12 (GOST 8239 - 72 Russian Standard).

Bending+Shear according S.Timoshenko

Wenbin Yu's picture

Dear Zishun and Sandyg,

In fact you don't have to assume constant shear over the beam height (equivalently section remains plane althoug not normal) to take advantage of the merits of the Timoshenko beam model. And the warping due to flexure contributes meaningfully to the shear energy. That is why we need a shear correction factor to compensate for this. A generalized Timoshenko model [1, 2] has been developed  by carrying out an asymptotical dimensional reduction from 3D elasticity equations to a 1D model in the form of Timoshenko theory. The Timoshenko model constructed this way can reproduce the flexure problem of the elasticity theory. And also there is no need to assume how the section should deform and no need to introduce shear correction factors. 

[1] Yu, W.; Hodges, D. H.; Volovoi, V. V.; and Cesnik, C. E. S.: "On Timoshenko-Like
Modeling of Initially Curved and Twisted Composite Beams," International
Journal of Solids and Structures,
Vol. 39, no. 19, 2002, pp. 5101-5121. (pdf)

[2] Yu, W. and Hodges, D. H.: "Elasticity Solutions versus Asymptotic Sectional
Analysis of Homogeneous, Isotropic, Prismatic Beams," Journal of Applied
vol. 71, no. 1, 2004, pp. 15-23. (pdf



Dear Dr. Zishun,


thanx for the reply. Can you plz tell me more about the thick and thin plate theory. I mean to say that lets say i have one thick and one thin plate so what will be the difference in approach to find the solution for deflection?



zishun liu's picture

Dear   Sandy,

For a thin plate, a classical thin (Kirchhoff) plate theory can be used (Pls refer to “Theory of Plate and Shells” by Timoshenko and Woinowsky-Krieger). When the thickness to length ratio of plate increases to 1/20 or higher, higher order plate theory of Mindlin and Reissner plate theory can been adopted to cater for the effect of transverse shear deformation. For theoretical solution, you also can refer to “Theory of Plate and Shells” by Timoshenko and Woinowsky-Krieger. If you want to find numerical solutions of thin and thick plates, the commercial software, such as ABAQUS, can be used to solve it.

Dear Dr. Zishun,

 Thanx a lot i will take that book.


Alan Tan's picture

    Dear Sandyg,

I did a research (for my masters) about thin and thick plates which also includes large deformations. I used FEM for that. I will try to find the FEM code. 





Dear Sir,

             1.  In FEM, I have not understood the exact difference between Implicit and Explicit techniques, though I have used both type of solvers.... Can u pls help me out in this?

              2. How to connect a beam to a shell element in FEM and a shell to a solid? All the three differ in their dof's.



Gerd Sebastiani's picture

Explicit vs Implicit Time Integration

The demand for a dynamic, time-dependent solution method introduces the need for time integration into the finite element setup. However, the choice of a time integration scheme may affect the quality of the obtained results, solution time, and numerical stability. Thus, a brief discussion of the main properties of the major time integration schemes is immanent.

In general, time integration subdivides the interval of the whole process time into finite subintervals Dt. Considering the whole problem as a linear dynamic system of finite element as given in eq. 1


where the matrices M, C and K correspond to the given mass-, damping-, and stiffness matrices of the model - the given schemes seek to solve eq. 1 at each finite subinterval. For doing so, two major schemes - namely the implicitexplicit scheme - are applied. Both schemes simplify eq. 1 to a form well known from static case, being dependent only on the new displacements as given in eq. 2 and



Here is D the effective stiffness matrix, incorporating the previously mentioned mass and damping matrices by means of appropriate integration constants. Likewise, the effective right hand load vector takes not only the current external forces on the system but also the dynamic contributions of the mass and stiffness matrices into consideration. Finally, the higher derivatives are interpolated based on the time step and the displacements.

The major difference between both schemes is given by the composition of D, the implicit scheme builds an effective stiffness matrix by use of the whole left hand side of eq. 1. The explicit scheme, on the other hand, takes only the dynamic contributions M and C along with the appropriate integration constants into account, while shifting the stiffness matrix to the right hand side. In addition, the explicit scheme evaluates the load vector based on the already known displacements at time t and t-Δt, the implicit scheme uses an additional guess for the reaction forces at time t+Δt. Since the latter are not known beforehand, a variational approach is used to solve an implicit system, whereas the explicit method can use a direct solver as well. as well as the evaluation of the effective right hand load vector. Considering the effective matrix

Introducing a variational approach results in special requirements on the condition of matrix D. An ill-conditioned system at least requires a large number of iteration to converge, if a solution can be obtained at all. On the other hand, the explicit approach will provide a solution as long as the time step is smaller than a certain threshold Dtcrit, given by the Courant-Levy condition in eq. 3



where le is the smallest element length in the model and ce the corresponding sonic speed within the element.

A minimum timestep restricted by geometrical and material properties will thus lead to high solution times for large systems consisting of small elements. While the computations can be artificially accelerated by a proper choice of the material dependent sonic speed, this so-called mass scaling will, on the other hand, introduce severe errors once the emphasis of a simulation is on the dynamics itself. Implicit solvers on the other hand are - once steady boundary conditions can be assumed - not bounded by a minimum timestep. In case of changing boundary conditions - such as a tool with a changing contact position - the timestep must be chosen sufficiently small to smoothly shift the contact from the initial to the next node along the toolpath.



K. Bathe: Finite Element Procedures, Prentice-Hall, Englewood Cliffs, USA, 2002, ISBN 81-203-1075-6

hi,i search for dynamic of composite beam matlab code

can u help me

ramdas chennamsetti's picture

R. Chennamsetti, Scientist, India

The difference -

If the semi-discretised equation ma+cv+kx=f(t), is written at 'i+1' time step to get 'x' at i+1 => Implicit

If it is written at 'i' time step to get 'x' at 'i+1' => Explicit.

[where a = acceleration, v=velocity and x=deflection/deformation, and v are expressed in terms of time derivatives of 'x'.]

In general 'x' at i+1 is a function of x at i, v at i etc in explicit, but, in implicit schemes 'x' at i+1 is a function of x at i, v at i+1 etc. => parameters in i+1 step are also appearing in the function.

In general, explicit schemes are conditionally stable and implicit schemes are unconditionally stable. But, there are some implicit schemes, which are conditionally stable.

Explicit schemes are used for shot duration phenomenon like shock loads, blast, impact etc. (high frequencies). Implicit schemes => long durarion   phenomenon (low frequencies).

You may refer 'Newmark's' time integration technique available in Structural dynamics books.

vh's picture


You can refer to the ABAQUS User Subroutines Reference Manual (e.g. at You will have to look for the documentation on the user subroutine UEL

V. Hegadekatte, University of Karlsruhe, Germany

Dear Sir,

Thank you very much for your suggestion !!!

Sai Sudha Ramesh

vh's picture


One can code the user element in Fortran or in C/C++. For writing the user element in C/C++ you can refer to the following webpage

V. Hegadekatte, University of Karlsruhe, Germany

Hi all,

I have visited quite a number of blog sites and such discussion boards, most of them full of trashy discussions--time wastage! However, in here, both the problems and comments are scholarly and pretty useful. I dont know if my problem is upto the mark or not. But please care to respond. Thanks  

I am working on modelling of composites. I took a 2D bitmap image of a composite, opened in matlab, converted it into pixel coordinates (file format: .mat) . Now the dimension of the matrix of pixel coordinates is around 20000x2. My objective is to export this image data into ABAQUS for diffusion analyses. Could anyone please let me how to import this huge image matrix into ABAQUS or create a geometry? I am thinking on converting or reformatting the .mat data file into abaqus .odb file. So the second question is, what is the format of .odb file that can be used in ABAQUS.  



The route that I have taken in the past is as follows:

  1. Assume that each pixel/voxel is one element.
  2. Arrange the pixel data into a format that you can easily convert into Abaqus format.
  3. Write a program to create an Abaqus input file from the pixel data.
  4. Create an odb file after reading in the Abaqus input file.

An example of such an approach can be found here.

In the example, I start off with a pixel file called <>.sub and then run createAbaqus to create the Abaqus input file <>.inp and the element and node files <>.ele and <>.nod.   The code generates ASCII files for 2D situations.   You can easily take the idea and apply it to your situation.  My work was done many years ago and used Abaqus 6.3.  For refinement you can also use createAbqRefine to refine the mesh by dividing each pixel into 4 elements - but not much more.

People who work with Abaqus on a regular basis will probably be able to provide a better solution. 

Dear Biswajit,

Thanks for your suggestion. Let me try this to implement to my problem first n then I'll get back in case I got stuck up.


ravitejk4u's picture

Ravitej Kommana
IIT Kanpur


i want the mid point deflection in case of "four point bend test" on a
specimen of length 140mm, width of 40mm and thickness of 10mm.

Problem is that, beam theory is valid if the cross-sectional
dimensions are less than axial dimensions. Here this condition is
violated. i had modeled the specimen in Abaqus and the diffection at
mid point is almost double that of the actual diffection calculated
from beam theory. 

how can i find the theoritical difflection at mid point in this case?

i had one option, using timoshenko beam theory and applying larger
deflection theory. will it be a reasonable approximation in this case?



Hi all

I have developped a finite element MATLAB program for a stepped beam

*with selectable number of portion of stepped beam and

*selectable size of beam.

The output of this program is the flap-wise and edge-wise natural frequency. Everything work perfectly.

My problem is that, i would like now to add the area moment of inertia of each portion of my stepped beam as a variable because the first portion of my stepped beam will be a "hollow beam".

I would like to find some interested information about MATLAB program of a stepped beam in order to improve the way i will input my data.

hope to get some advice and link soon!


lagouge TARTIBU


Mechanical Engineering

Cape Peninsula University of Technology

Cape Town

South Africa

lagouge TARTIBU

Researcher Mechanical Engineering

Cape Peninsula University of Technology

Cape Town

South Africa

I am trying to solve a vibration problem of timoshenko beams by considering the effect of geometric non-linearity with von-karman type strain-displacement relations. I have three variables which are theta, u and w. I am getting correct linear value but non-linear some thing wrong with energy expression it seems.

Can any body suggests some tips to write the strain energy of timoshenko beams due to axial, shear and bending loads considering the geomentric nonlinearity of von-karman type?  Looking forward to your valuable comments.






Wenbin Yu's picture


I will suggest you to read Nonlinear Composite Beam Theory by Prof.Hodges at Georgia Tech

You will get ideas how to write the energy expression and how to handle large deflection correctly.





Is there a book that i can check for the stiffness matrix for vibration of beam allowing for effect of slip..?

Hi to all! I am new to this forum and I just want to ask if there is a difference on the mathematical description of a horizontally and vertically configured cantilever.  i am doing a research which requires vibration measurment and analysis, and I used a hacksaw blade and strain gages to do these.  The hacksaw blade is coupled to the vibrating body, and the vibration is measured via the strain gages mounted on the hacksaw blade which is positioned vertically.


I hope somebody would answer this question.  Thanks in advance! 

Dear all,

i have a problem in FE and i hope to help me in it. i make FE analysis for my master research work in ansys. i have two prestressed concrete beams provided with dry joint in the mid span, the first beam is a control beam ( without additoinals), the second one is strengthened using FRP strip along the bottom face of the beam and u-wrap distributed by spacing along the beam. i did initially debonding for the FRP strip at the mid part of the span= 900 mm ( that means, i merged the joints of the FRP strip with the joints of concrete beam at the same location except at the part of the mid span-initially debonding- to avoid stress concentration. but i faced a problem, that the deflection for the first beam and the second beam are the same however, the second one should be lower in the deflection due to using FRP and the strains in u-wraps are almost zero. i used shell 99 for frp strips and u wraps and solid65 for concrete elements and link 8 for prestressing. could you please help me.

also there is a factor in the real constant of shell 99 named " elastic foundation stiffness" i put it equal to zero but i do not know what is that mean and if that effective what is the amount i have to use.

 really many thanks



Dear baltkobe

I am interested in using beam element in abaqus and define the stress
resultant- displacement nonlinear curve ( Axial-displacement,
Moment-curvature, torque-twist angle), could you please guid me , how i
can define these curves for beam behaviour?

It's my e-mail adress:

Your reply would appreciated in advance.


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