Literature Review

In the I-beam section, global bending would often occur in the XYZ planes just like in the original rectangular section. The universal bending equation is given from the slope and deflection formula under the facilitation of methods such as double integration, energy methods, Castiglione’s theorem and the method of sections. In Castiglione’s method, the techniques extend to include analysis of indeterminate structures for purposes of bending. In 1 D bending analysis, a solid orthotropic beam is singularly analyzed using the common methods. However, the same concept can also be extended to complex 3D beam analysis. Normally, the beam is made up of multiple orthotropic solids hence can be configured as separate entities during analysis with each having a thickness of ‘H”. Typically, the I-beam will have three major sections (as far as bending and stress analysis are concerned); namely; the web, top and bottom parts. Therefore, structural analysis entails determination of the principle design parameters such as shear and moment of area Ixx, Iyy and Izz and therefore bending stresses are determined as we have to consider various loadings and stresses in all directions.

For determination of the beam deflection, the superposition method often comes in handy when analyzing and designing beams against deflection in either static or dynamic scenarios. For static beams, there are beams at rest which have completely different analytical techniques from the dynamic beams. Therefore deflection in beams could depend on the type of loading. Macaulay function is often used to represent distributed loadings. As the loading becomes more complex, such that there is combined loading in the lower and upward directions the principle of superposition often come into play as illustrated in the diagrams below.

Cheng (1979) proposed various techniques of analyzing the orthotropic beams. He dwelt on orthotropic beams by analysis through the application of infinite series. However, the situations are out of scope for the discontinuities. The beam was assumed to be constituted of orthotropic solids having edges parallel to perpendicular axes of elastic symmetry. However, the thickness of beam was assumed negligible such that analysis only considered plane stresses. Notably, the equations of compatibility were observed:

D⁴?/dx⁴+2k D⁴?/dx⁴n + D⁴?/dx⁴ = 0

Where K= (ExEy)^0.5/2(1/Gxy-2Uxy/Ex)

As the analysis is extended to deflection of beams, the above equations are changed such that the new equations are:

Ex U= ∫ (?x-Uxy?y)dx+r(y)

Ey U= ∫ (?y-Uyx?x)dy+s(x)

These equations above will considered deflection both in the x and y directions. The principle of superposition could also be applied to determine the combined bending in both X and Y directions. As mentioned earlier, although this method can be efficient for infinite conditions, discontinuities are not entertained by the method. Pathak and Patal (2013) also dwelt on a similar method although they used the flexural members. The techniques can also be extended to analyze the orthotropic beams. According to the authors, the stress-strain relations for isotropic material are:

?x = C11?x +d12?y ; ?y= C12?x+ C22?y

For small displacements, the relations between strain and stress can  be:

Method of Orthotropic Beams Bending Analysis

 ?x=du/dx;             ?y=dv/dy and such solutions of such nature can be solved by considering the following matrix forms:

d/dy          u                           0       -a     0         1/G               U

                 v         =                c1a    0      c₂             0                 v

                 y                          0        0       0          -α               y

                 x                            cα²G 0      c1α        0                x

The methods are used to analyze beams under symmetric central loading. Therefore, the method is known as method of initial functions (MIF). The equations used are originating from the Hooke’s law and the equations of equilibrium. For analysis of free vibration in rectangular sections, the values of frequencies are analyzed using the Timoshenko beam theory (US Department of Agriculture, 1979). The method is only restricted to analysis of beams for stresses and deflections.

For a real time analysis, Srikanth and Kumar (2016) proposed a simulation method. The method picks up from where classical theory of plates scoped. CTP often assumed plane is orthogonal before and after stresses and bending applications. However, the degree of accuracy in application of such methods in orthotropic beams is often decreased.  

Before we can determine the bending stresses, it is often imperative to analyze various parameters of the beam which will lead to determination of the designated parameters.

Step by step Procedure

In determining the beam stresses and bending moment and deflection, we can analyze in the following step by step procedure:

Consider cantilever orthotropic beam being subjected to bending stress under single force application.

According to Aktas (2016), the stress component is given by:

U= -P/I [al1/2x²y +a 16/Ia11{b²+12y²}/24 x] + f(y)

Hence the first step is to determine the second moment of area substituting in the formula:

 I= hb³/12 where h= the thickness of beam plates and b is the breadth of the beam.

Suppose the loads are distributed then the bending stresses are given as:

?= qx²y/2l+q/h[a16/a11x/b(1-12y²/b²)+2(2a12+a66/4a11-a²16/a²)(4y³/b³-4y/5b)]

And the deflection function would be determined from:

?= q/2h(-1+3y/b-4y³/b³)

For the above case, the iteration procedure can be applied such that:

First we determine all the elements in the equation such a11 and  a66. Then we substitute in the bending stress equation above.

To further advance the idea, the deflection and bending stresses could be obtained by considering the following steps:

1. Firstly, obtain the equivalent modulus and strength values such as : α, β and γ for unidirectional, fabric and transverse orientations. Also obtaine the stiffness and material strength values
2. Secondly, determine the beam stiffness coefficient and this is given by: D= ½(Ex)f. tfb²wbf+ ½(Ex)wtwbw³+1/6(Ex)f. t³fbf  where F= (Gxy)wW for both bending and shear stresses
3. Now, determine the deflection for each plate orientation, be it  XY, YZ or XZ planes using the formula: δmx= δb +δs = PL³/48D + PL/4Kf    where Ky= 0.9

Now, due to bending, the beam deflection is : δb= PL³/48D and Due to shear: δs=PL/4KyF

1. Determine the appropriate stresses and check for maximum bending shear in all the plate orientations

The approximate maximum bending and shear stresses can be obtained from the equations: ?x= ExEs and τxy= Gxyγxy

For the purpose of design and based on the requirements and expectations always design with a factor of safety being incorporated

1. The beam global buckling load

For critical buckling of beam, always substitute in the equations below:

Pcr= 17.17/L²(D.JG)^0.5(1+π²Iww/L²JG

Where JG= 2(Gxy)t³fbf/3+ (Gxy)wb²wbw/3 and Iww= (Ex)ftfb²wb³f+ (Ex)ft³fb³f/36+ (Ex)wbw³/144 and the F.o.S is to be 2.36

1. After the critical elements have been determined for different plates, it is now time to extend it further for complex analysis. Initially, we were working on individual elements but now we must merge into a single case using the techniques analyzed in the literature review above
   • Firstly, due to deflection, the maximum deflection is obtained: W(x,y)=

This equation can be twisted slightly to have: . This is the universal deflection equation (as it considers all deflection in all directions)

To further minimize we must obtain the eigen problem below: 

And λ=phw²a²b²     and δis= {1 for r=s and 0 for r/s

This elements can be represented in the matrix below:

Q1                     K11        K12      K13     K14        

Q2                     K21        K22      K23     K24         Q11

Q3         =          K31        K32      K33     K34         Q22

Q4                     K41        K42      K43     K44         Q12

Q5                     K51        K52      K53     K54         Q66

Q6                     K61        K62      K63      K64 

Where    D1                    Q11

                D2    = h³/12    Q22

                D3                    Q12

                D4                    Q66

Briefly, consider the following :

-always provide a near perfect estimation of Q11, Q22, Q12, Q66 and D1 to D4 using the equations provided

-Form and solve eigen value

-Identify mode and find Hr

-Form matrix solution an solve

-Does it converge? If yes, then design is safe otherwise repeat the whole procedure by considering the limits of convergence from the raw values.  

It is clear that the above techniques are used in plane stress models of orthotropic plates. Normally, as mentioned earlier, the plates are to be handled separately and then superposition principle can be used to merge them. So in a nutshell, we determine the stresses in the coordinates X-Y, X-Z and Y-Z individually. In 2D it was assumed that the plane remains orthogonal throughout such that the common analytical techniques can surface. However, as the solids become more complex, as in the case above, the assumption becomes superfluous an therefore the above techniques comes in handy. However, it should be noted that the methods presented only works in scenario where the principle stresses are such that the infinite series can smoothly be applied.

Aktas, Alaatin. Determination of The Deflection Function of A composite Cantilver Beam Using Theory of Anisotropic Elasticity. Kirikale. 2016.

Cheng, S & Liu,JY. Analysis of Orthotropic Beams. 1979

Rakesh, Patel, Dubey, SK & Pathak, KK. Analysis of Flexural Members Using an Alternative Approach. 2013

US Department of Agriculture. Analysis of Orthotropic Beams. Madison Winconsin. 1^st (ed).1979