Mathematical Analysis And Matlab Modeling Report Assignment Sample

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Introduction - Analysis of Shear Force and Bending Moment in Spring-Supported Slabs and Conventional Beams

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TASK ONE

1.1 Problem definition

The problem is based on the analysis of shear force and bending moment of a slab supported below with the help of springs of stiffness “K”. The properties of the slab are given in the problem statement such as the length of the slab is “L” and having a thickness of “t”. The slab is connected at one end. There will be differential equations related to the analysis of the slab. The slab is given a uniformly distributed load of “q”. The uniformly distributed load is spread all over the slab. There will be differential equations related to the analysis of the parameters of the slab. Diagrams have to be generated on the shear force and a bending moment of the slab. There will be a second case and that has to be considered for the analysis purpose. The second case will be the same as the first case except for the ground springs attached to the slab. The slab will be considered as a conventional beam hanging in the air and attached at one end without the support from the ground springs as in the first case.

The Ordinary Differential Equations have to be developed for both the first case and the second case. The ordinary differential equations have to be solved without considering the ground reaction from the springs connected by applying the boundary conditions. The thickness of the slab has to be considered to get a maximum deflection of 5mm for the conventional beam and the same procedure has to be applied to get the results for the slab which is supported in the ground with the help of springs. The results of the analysis have to be differentiated for comparison. The ordinary differential equations have to be solved with the help of a different stencil and the process has to be continued to get a satisfactory result on the differential equations. The equations generated have to be formulated on Matlab to plot the results and deflections. The results have to be compared with the conventional beam. The distribution of the soil reaction has to be shown with the help of deflections. The contact stress value, resultant soil reaction force, and its point of application have to be determined using a suitable numerical integration method. The sheer force and bending moment diagram have to be determined by applying suitable numerical analysis from the known deflection.

1.2 Mathematical analysis

It has to be done to find out different parameters in the problem statement. In sub-task 1, the mathematical tools that are used in the analysis have been discussed. In sub-task 2, the results from the analysis have been discussed (Singh et al. 2020).

1.2.1 Sub-task 1 100

The different parameters have been considered for both the tasks such as the concrete slab supported in the ground by springs of fixed stiffness and a conventional beam not supported in the ground with a uniformly distributed load on both of them. There will be two different differential equations used to find out the reaction force from the ground. The total reaction will be the summation of the reaction from a uniformly distributed load and the load from the hinge point (Macedo et al. 2020).

The differential equations considered for the slab with uniformly distributed load is

“F_udl = w*[-L/2;(L^2)/48-(L^2/48);(L^2/48)];”

“F_pl = wp*[-1;-0.05;0];”

And the differential equations considered for the conventional beam with uniformly distributed load is

“F_udl = w*[L/2;(L^2)/48-(L^2/48);(L^2/48)];”

“F_pl = wp*[1;-0.05;0];”.

The following equations can be considered to observe the parameters that are considered for the sub-tasks.

1.2.2 Sub-task 2

The analysis of shear force and the bending moment has been done with the help of these differential equations. The reaction force from the ground will be the summation of these two forces. The reaction force will be different for both the cases of the concrete slab and the conventional beam (Zhang et al. 2020). The equation for the calculation of reaction force on the concrete slab is

“Reaction = [Rc(1)+w*L/4; Rc(2)+w*(L/2)^2/12; Rc(5)+w*L/4];”

And the equation for calculation of reaction force for conventional beam is

“Reaction = [Rc(1)-w*L/4; Rc(2)+w*(L/2)^2/12; Rc(5)-w*L/4];”.

The following equations can be considered for calculating the reaction force for both the cases mentioned in the sub-task.

1.3 Modeling methodology & solutions

The modeling of the problem has been done by considering the above differential equations. In this part of this task several mathematical analysis has been configured with the help of the several mathematical tools and respective associated formulas (Qi et al. 2020). According to the several formulas different analytical parts are extracted. The mathematical formulas have been considered from several theoretical aspects and respective data has been conquered according to the size of the cantilever beam with spring support. After the configuration of the mathematical formulations, those are assigned into the Matlab and respective results have been obtained.

According to the above article figure several formulas and data could be visualized and respective values are also assigned and respective associated different analysis has been illustrated in the below sections (Yu et al. 2020). Based on the software simulations several results and the deformation of the beam could be easily observed.

1.4 Results and discussion

The results have been obtained for the analysis. The value of the Reaction force and displacement has been found. The displacement is due to the pressure generated from uniformly distributed load (Salvy et al. 2019).

1.5 Conclusions

The project has been executed taking into account the problem statement and the analysis part has been done considering two things and they are concrete slab and conventional beam. There are different parameters that are found for analysis such as the sheer force diagram and the bending moment diagram. To calculate the sheer force and the bending moment, ordinary differential equations have to be generated. The solutions to the ordinary differential equations have been found using different numerical methods and techniques. The reaction from the ground for the concrete slab has been found and the resultant reaction force and the point of application of the resultant reaction force have also been found. The reaction force is due to the springs attached to the ground with the concrete slab. The sheer force diagram and the bending moment diagram have been generated for the concrete slab and the parameters that are taken into consideration are the length of the slab, the thickness of the slab, the ground reaction force from the springs attached to the slab with fixed stiffness, and the value of the uniformly distributed load. Different analytical methods have been adopted to solve the ordinary differential equation. The equations generated have been imported into Matlab using suitable codes. The equations have been solved in Matlab and the diagram for shear force and the bending moment has been plotted.

TASK TWO

1.1 Problem definition

Task 2 of the project has to be executed considering the Conventional beam into consideration. The conventional beam has to be analyzed for the sheer force and bending moment. The ordinary differential equations have to be generated for finding out the sheer force and bending moment. The Ordinary Differential Equations have to be solved with the help of Finite Differences Stencil and this process has to be continued to get a satisfactory result for the equations for the unsupported beam. Responsive several software simulated results had to be observed and respective various deformation of the beam has been observed.

1.2 Mathematical analysis

The analysis has been done on the Matlab platform. The Ordinary Differential Equations generated are solved in Matlab. The value of shear force and the bending moment has been found using the Differential Equations (Yang et al. 2021).

1.2.1 Sub-task 1

The conventional beam is divided into several parts. A small part of that has been considered and forces acting on the small part have been found. The small part considered has been integrated from zero to the length of the conventional beam to find out the total forces acting on it (Liu et al. 2021).

1.2.2 Sub-task 2

The calculations are made taking the differential equations into account. The value of shear force and the bending moment has been found. Matlab has been used to determine graphs of Shear force and bending moment (Guo et al. 2018).

1.3 Modeling methodology & solutions 200

As per the above-attached figures several relevant issues and re-elected several systems have been designed in the software “MATLAB” on the basis of different simulations and respectively associated analysis has been considered and respective several issues has been observed (Aranda et al. 2022).

On the basis of the several related aspects and associated formulas have been assigned in the software and associated different analytical methods are applied. Moreover as per the above attached system and respective associated several formula and related aspects are implemented. Moreover, several related data and the results are obtained (Trujillo et al. 2018).

1.4 Results and discussion

In this section several results and related figures generated from the software have been illustrated and several results have been described. According to the different deformation and associated systems, several systems have been observed and various results have been obtained. Various system and associated values have been put first and respective analysis has been elaborated (Jalalian et al. 2019).

Based on the several assigned values of the unsupported beam the respective deformation and the plots have been obtained.

This figure elaborates the several systems and respective different relevant plots, as per the several plots respective associated analysis has been elaborated.

The above-attached figure illustrates the different results retrieved from the software analysis of the bam and associated features.

The above figure demonstrates the deformation and the respective bending moments and shear forces that could be observed. Moreover, several related issues and the several data provided with the cantilever beam have been implemented. Considering the obtained results the analysis and the strength of the beam could be easily observed.

1.5 Conclusions

The task has been completed taking into consideration the conventional beam that is used for the purpose of construction of buildings and bridges. The task is to find out the value of the sheer force and bending moment of the conventional beam. Ordinary Differential Equations have been generated to analyze the forces on the beam. A small part of the beam has been considered to check the forces acting on that part. Method of integration has been applied to calculate the total forces acting on the whole part of the conventional beam. Ordinary Differential Equations have been generated by this method. The equations generated are solved using a difference stencil and the process has been repeated till a satisfactory result has been obtained for the equations. The sheer force diagram and bending moment diagram has been obtained from the equations. The process has been established in Matlab. Codes are generated to solve the differential equations easily. The sheer force diagram and bending moment diagram has been plotted in the Matlab platform.