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:
?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:
Now, due to bending, the beam deflection is : δb= PL3/48D and Due to shear: δs=PL/4KyF
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
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
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. 1st (ed).1979
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