FEA (LUSAS SOFTWARE)
The free body diagram is clearly shown below:
There will be two forces acting against the force F; these are forces Fx and Fy.
Assuming a perfect rectangle as shown below, and then dividing it by 2:
From the similarity of force as well as relative dimension triangles
But according to the Rankine formula, F is gotten by:
From the above equation, F is the vertical force acting on the cantilever, E is modulus of a given elasticity, I is area for moment of inertia, L is the unsupported length of column, and K is the column effective length factor.
K for this case will be 2.0 since the cantilever is fixed at one end while the other end is free:
And the area of moment is;
2) Failure will be prone at the point where stress is maximum.
maximum stress M will be found by: (m*C)/I
From the above equation, m is the provided moment, I is the area for moment of a given inertia for the beam, while C is the distance from required centerline to the given point where the force is acting.
Moment m = F*L = 19.369 *5kNm
M = (96.845*5)/6.54
M = 74.0405kN/m
Therefore, failure of the cantilever will be prone at the point where stress is 74.0405kN/m. this is obviously at the point of where the cantilever is fixed.
With the addition of the steel bar, the moment of inertia of the beam will change and hence point of failure and magnitude of force
Hence I = 6.55m
M = (96.845*5)/6.55
The total length for the steel bar is 5m.
To check for the best ratio, cantilever end deflection is checked for both cases:
Where deflection is given by 3(σu,t)(1-v)/E (L/T)^2
For t= ds1 and for t=ds2
For ds1=2m the deflection is 3.75
For ds1=3m the deflection is 3.89
For ds2 =2m the deflection is 3.90
For ds2=3m the deflection is 3.76
Therefore, the best ratio of ds1/ds2 is 3:2
In comparison to the case in question 4, the magnitude of force is increased due to the increase in thickness t hence change of failure point.
FINITE ELEMENT METHOD (FEM)
FEM can be termed as the numerical technique for obtaining approximate solutions to problems (boundary value) for differential equations. It makes use of variation methods, for example, the calculus of variations to reduce an error function and subsequently generate a stable solution. It is similar to the concept of joining together many tiny straight lines to approximately create a large circle and it encompasses all known methods for joining many simple element equations together over numerous small sub domains known as finite elements, to approximately determine more complex equations over a much larger domain. FEM has grown pretty rapidly to become the most essential numerical analysis tool for applied mathematicians and engineers because of its obvious advantages over the previous methods that were in use. The biggest merit that it has had been its applicability to arbitrary shapes of any number of dimensions. Moreover, the shape under study can be made of non-homogenous material or/and anisotropic material. The manner in which the shapes are restrained or fixed can be general and so can the applied sources. The applied sources include pressure, force, and heat flux among other things. The FEM brings forth a standard process for converting governing differential equations or governing energy principles into a multiplicity of matrix equations to be evaluated for approximate solutions. These solutions are very quick and accurate for linear problems. After getting an approximate answer, the Finite Element Method gives additional standard procedures for the calculations that follow after (post-processing) for example determining the derivative or integral of the gotten solution or at a specific point on the shape.
When the Finite Element Method is used in a given field of analysis for instance thermal analysis, vibration analysis, and stress analysis, it is referred to as Finite Element Analysis (FEA). An FEA is one of the most common tools for structural and stress analysis. Most fields of study are usually related. For example, the distribution of non-uniform temperature may induce non-obvious and unexpected loading conditions on solid structural members. Therefore, it is easy to conduct a thermal finite element analysis to obtain temperature solutions that in turn work as the input data for stress FEA. FEA has the capability of receiving data from other tools like the computation fluid dynamics systems and the kinetic motion analysis systems.
Basic Integral Formulations
The foundational concept behind the Finite Element Method is replacing all complex shapes with the summation (union) of large numbers of very basic and simple shapes like triangles that are put together to accurately model the original part. The basic, smaller, and simpler shapes are known as finite elements because every one of them occupies a small and specific finite sub-domain of the original part. The finite elements vary to an infinitesimal level and have been used for centuries to derive differential equations. To give a simple and straightforward example of the summing and dividing process, consider the determination of the area of an arbitrary shape below.
With the knowledge of the equations of the bounding curves, it is possible to integrate and get the enclosed area. Otherwise, one can divide the area into an enclosed collection of triangles that are covering the given shape with a mesh. Afterwards, the summation of the areas of the individual triangles can be done:
There are a couple of choices to be made concerning the triangles desired. One could either go with the cubic triangles, quadratic triangles (usually have edges that are parabolic), or the linear (straight sided) triangles. The area of a linear triangle is given by a simple algebraic expression. After numbering the three vertices in an anticlockwise order, the area is given by;
Ae = [x1(y2-y3) + x2(y3-y1) + x3(y1-y2)]/2 and the triangle’s centroid is located at Xecg =[x1+x2+x3]/3, Yecg = [y1+y2+y3]/3
Similar expressions provide the moment of inertia components. Therefore, one has to gather (extract) the given element vertices coordinates from a set of mesh data for use in the computation of the area of a linear/straight side triangle. It is comparatively easy to calculate the area of the curved triangle by use of the numerical integration. It is, however, computationally more expensive to obtain the area of the curved triangle than it is to obtain the area of the linear triangle. It is vital to note that the linear or straight-sided triangle mesh, estimates the area better than the other two methods. It, however, introduces geometric errors along the curved boundary. This boundary geometric error in a straight sided triangle mesh comes from replacing a boundary curve by a number of straight line segments. The error (geometric error) in straight sided triangles can be minimized to any preferred level by increasing the total number of linear triangles. That decision, however, increases the number of manipulations and calculations, and makes one trade off the sum of preferred area summations and calculations versus geometric accuracy.
Area is a scalar quantity, so it is within acceptable rules to sum its parts to find the total value as demonstrated. Other physical quantities like for instance strain energy, mechanical work, and kinetic energy can be added up in the same fashion. The Finite Element Method always involves a couple of governing integral statements that are converted to a matrix system by the assumption of how items differ within a typical element. That integration is equally converted to the total sum of integrals over all elements in the mesh. Starting with a governing differential equation still converts it to an equivalent integral formulation by the MWR (methods of weighted residuals). There are two very common weighted residual methods for finite element analysis and these are the Method of Least Squares and the Galerkin method.
Gather and Scatter Operators
An integral evaluation for a Finite Element Analysis requires a mesh. Characteristically, it is a triangular mesh designed for surfaces and with a tetrahedral mesh for solids. The outcome of a finite element mesh generation produces at least two sets of data. The first set of data (nodal data) is the numbered list of each generated vertex together with their spatial coordinates. The second set of data is an element set and is the numbered set of elements along with the numbered list of element vertex numbers to which it is connected. This is commonly known as the element connectivity list. The connectivity list is the vital data that allows the Finite Element Analysis calculations to be automated. All FEA use operations that deal with vertex numbers (specific node numbers) of a single element.
The two operations carried out are usually referred to as gather and scatter or assembly operations. The gather operation is useful in bringing the known nodal data found in the full mesh back to a unitary element. The velocities and coordinates used in the element integrals above were thought to be stored with the mesh nodal data. While the mesh may have a significant number of nodes, every linear triangle element only has three nodes. The gather operator made use of the element connection list to get the data for a given element in the summation to get its three nodal velocities. The assembly or scatter operation is the reverse of the gather operation. It is said to be the partial summation of element data to the given matrices associated with the mesh data. A scatter gets something related to the local nodes of an element and sums them with the corresponding matrix item at the full mesh level.
Geometric Boundary Errors
One may imagine that the geometric boundary error stated for the linear triangle is done away with by choosing to make use of the mesh of curved quadratic triangles. The parabola segments go through three points located exactly on the boundary curve but can easily degenerate to straight lines if in the interior. Therefore, the boundary shape error is minimized at the expense of more complex area calculations, but not done away with completely. Some geometric errors are maintained because most engineering curves comprise of circular arcs, nubs (non-uniform rational B-spines), or spines and, therefore, are not matched by a parabola. The commonest way to minimize mesh geometric error is to make use of numerous smaller elements. The default element selection in solid works (SW) Simulation is without doubt the quadratic.
Zienkiewicz, O. C., & Taylor, R. L. (2009). The finite element method for solid and structural mechanics (7th ed.). Amsterdam: Elsevier Butterworth-Heinemann.