Structural Analysis for Comparison of Bending of Beams

AIM:

To determine the Deformation, Stress, and Strain when Uniformly Distributed load is applied on Cantilever beam of Rectangular, I and C Cross-section, to run a Grid dependency test, to verify results using Analytical calculations using MATLAB and to select the beam based on the results.

DESCRIPTION AND GOVERNING EQUATIONS:

The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets of mechanics of materials. A beam is a member subjected to loads applied transverse to the long dimension, causing the member to bend.

Some of the physical examples of Cantilever beam subjected to UDL are Aircraft Wing, Bridges, Truss, Cranes, etc.

The wings of the airplane are cantilever For Cantilever Cranes

Governing Equations: Cantilever Beam

Consider a cantilever beam where one end is fixed and another end is free.

The loading condition is uniformly distributed load

Load P = W*x (N) acting at a certain distance x is

Stress at Each Point = W*x/A (N/m^2)

Strain at Each point = Stress * E

Deflection at Each point = `W/24*(-x^4+4L^3 x-3L^4)`

At x=0 `δmax=(WL^3)/(8EI)`

At x=L `δmax=0`

Stress due to bending of the beams

`σ_b=(M*y)/I`

M is the bending moment N-mm

`σ_b` = Bending Stress in N/mm^2 or MPa

y is the distance of the fiber top fiber from the neutral axis in mm

I is the second-moment area about the neutral axis in mmˆ4

The above equations are used to find the Stress, Strain, and Displacements for the beam.

OBJECTIVE:

1) To run a baseline Simulation and to perform a Grid dependency test.

2) To determine the Deformation, Stress, and Strain when Uniformly Distributed load is applied on the Cantilever beam of Rectangular, I and C Cross-section and to select the beam based on the results.

3) To verify results of SOLIDWORKS SIMULATION with analytical calculations using MATLAB.

PROCEDURE:

The Entire Process of FEA analysis is divided into 3 parts

1) Pre-Processing

Geometry Creation

Generation of Mesh

Assignment of Material properties

Boundary Conditions

2) Processing

Governing Equations are applied to the discretized elements and displacements will be calculated.

[F]=[K]*[X]

F=Loads

K= indicates geometric and material properties

X=Displacement

Run the simulation

3) Post-Processing

After the calculation of the displacements across the nodes the Stress and Strains can be calculated at any point.

Create the required Contours and Plots

Interpret the Results

Modify the geometry to get the optimized results

PRE-PROCESSING:

GEOMETRY AND MATERIAL PROPERTIES:

All beams are assigned Alloy steel

Properties: Alloy-Steel

CASE 1: RECTANGULAR BEAM

CASE 2: I BEAM

CASE 3: C BEAM

BOUNDARY CONDITIONS:

Here all the beams are Applied by uniformly distributed load of 1500N on the Smaller top face (i.e. 45mm face for case 1, 40mm face for case 2 and 45mm face for case 3)

And is Fixed from one end

CASE 1: RECTANGULAR BEAM

CASE 2: I SECTION BEAM

CASE 3: C SECTION BEAM

GRID DEPENDENCY TEST:

After performing a baseline simulation we will be performing Grid dependency test to select the mesh for further simulation

CASE 1: RECTANGULAR BEAM

Here the global mesh is selected as an input variable with size varying from 2mm to 16mm of grid size with an increment of 1mm

Stress is selected as a constraint.

Results of the grid dependency test:

As we can see that as the mesh gets refined the amount of stress values changes and we will be using the 3mm grid size for further analysis of Rectangular beam as it is best in terms of results and the computational time required will also be less.

CASE 2: I SECTION BEAM

Here the global mesh is selected as an input variable with size varying from 2mm to 10mm of grid size with an increment of 0.5mm

Stress is selected as a constraint.

Results of the grid dependency test:

As we can see that as the mesh gets refined the amount of stress values changes and we will be using the 2.5mm grid size for further analysis of I beam as it is best in terms of results and the computational time required will also be less.

CASE 3: C SECTION BEAM

Here the global mesh is selected as an input variable with size varying from 2mm to 10mm of grid size with an increment of 0.5mm

Stress is selected as a constraint.

Results of the grid dependency test:

As we can see that as the mesh gets refined the amount of stress values changes and we will be using the 2.5mm grid size for further analysis of C beam as it is best in terms of results and the computational time required will also be less.

MESH:

CASE 1: RECTANGULAR BEAM

CASE 2: I SECTION BEAM

CASE 3: C SECTION BEAM

PROCESSING:

HIT RUN

Governing Equations are applied to the discretized elements and displacements will be calculated.

[F]=[K]*[X]

F=Loads

K= indicates geometric and material properties

X=Displacement

POST-PROCESSING:

RESULTS AND CONTOURS

CASE 1: RECTANGULAR BEAM

STRESS CONTOUR:

Here we can see that he as the uniform distributed load is applied at the top, after bending the top fibers undergo tension and bottom fibers undergo compression but the magnitude remains same and the Maximum stress or Resistance occurs at the fixed end of the top and bottom fibers of 11.859 MPa

And the central fibers undergo neither tension nor compression hence are subjected to the stress of 0Mpa

DEFORMATION CONTOUR:

The maximum deformation occurs at the Free end of magnitude 0.03104mm

STRAIN CONTOUR:

Maximum strain occurs at the fixed end of top and bottom fibers of magnitude 3.757E-05

and Minimum stress occurs at the central fibers

CASE 2: I SECTION BEAM

STRESS CONTOUR:

Here we can see that he as the uniform distributed load is applied at the top, after bending the top fibers undergo tension and bottom fibers undergo compression but the magnitude remains same and the Maximum stress or Resistance occurs at the fixed end of the top and bottom fibers of 28.317 MPa

And the central fibers undergo neither tension nor compression hence is subjected to the stress of 0Mpa

DEFORMATION CONTOUR:

The maximum deformation occurs at the Free end of magnitude 0.0858mm

STRAIN CONTOUR:

Maximum strain occurs at the fixed end of top and bottom fibers of magnitude 9.636E-05

and Minimum stress occurs at the central fibers.

CASE 3: C SECTION BEAM

STRESS CONTOUR:

Here we can see that he as the uniform distributed load is applied at the top, after bending the top fibers undergo tension and bottom fibers undergo compression but the magnitude remains same and the Maximum stress or Resistance occurs at the fixed end of the top and bottom fibers of 55.734 MPa

Here the C section beam is subjected to torsion as we can see the twisting of the beam

DEFORMATION CONTOUR:

The maximum deformation occurs at the Free end of magnitude 0.03037mm

STRAIN CONTOUR:

Maximum strain occurs at the fixed end of top and bottom fibers of magnitude 1.840E-45

and Minimum stress occurs at the central fibers.

ANALYTICAL METHOD USING MATLAB CODE:

In this analytical calculation is done using MATLAB for Rectangular and I section beam.

We will be using the formulas mentioned in the description and governing equations above.

Using the above formulas

Moment of inertia for Rectangular Section:

Ixx=(45*80^3)/12

(Ixx)rect=1920000mm^4

Moment of inertia for I Section:

Ixx=[(40*(80)^3)/12-(36*(66.83)^3)/12]

(Ixx)I=811228.42

CODE:

% MOHD AZHAR MERCHANT
% BENDING AND DEFLECTION OF CANTILVER BEAMS WITH RECTANGULAR AND I CROSS-SECTIONS

clear all 
close all
clc

l=400;             %length in mm
E=210000;          %elastic limit in MPa
P=1500;            %load applied uniformly in N

y=80/2;            %distance from neutral axis for bending for all cross-section
M=P*l;             %Moment due to udl for all cross-section N-mm
 
%BENDING STRESS IN DIFFERNET CROSS-SECTION

%MOMNET OF INERTIA FOR RECTANGULAR AND I CROSS SECTION
I= [1920000 811228.42]

for i=1:2
   Bending_Stress(i)= (M*y)/(I(i));        %Bending Stress in Mpa
   
   Deflection(i)=(P*(l)^3)/(8*E*I(i));     %Deflection in mm
   
end

Bending_Stress_Rectangular=Bending_Stress(1)
Deflection_Rectangular=Deflection(1)

Bending_Stress_I=Bending_Stress(2)
Deflection_I=Deflection(2)

RESULTS:

CONCLUSION:

COMPARISON OF BEAMS

BEAMS	MAXIMUM STRESS (MPa)	MAXIMUM DEFORMATION (mm)	MAXIMUM STRAIN
RECTANGULAR SECTION	11.859	0.03104	3.757E-05
I SECTION	28.317	0.0858	9.636E-05
C SECTION	55.734	0.3037	1.840E-4

COMPARISON SOLIDWORKS SIMULATION WITH MATLAB:

	SOLIDWORKS SIMULATION		ANALYTICAL SOLN USING MATLAB
	MAXIMUM STRESS (MPa)	MAXIMUM DEFORMATION (mm)	MAXIMUM STRESS (MPa)	MAXIMUM DEFORMATION (mm)
RECTANGULAR SECTION	11.859	0.03104	12.5	0.0298
I SECTION	28.317	0.0858	29.5848	0.0704

From the comparison of Stress contours, the maximum stress is least in case of the rectangular section, intermediate in case of I section and largest in case of the C section of beam.

From the comparison of Strain contours, the maximum strain is least in case of the rectangular section, intermediate in case of I section and largest in case of the C section of beam.

From the comparison of Deformation contours, the maximum deformation is least in case of the rectangular section, intermediate in case of I section, and largest in case of the C section of beam.

We can see that all the beams are subjected to1500 N of UDL have their stress values way below the yield strength hence none of them will fail under the given condition but we can observe that the C Section undergoes torsion which is undesirable in case of beams hence we would least prefer it from the 3 beams.

From the Stress contours of Rectangular and I section beams we can observe that the Stress generated is more and uniformly distributed in the case of I compared to that of the rectangular section and hence I beam is most preferred compared to rest of the beams.

From the comparison of the results obtained, we can observe that results obtained from SOLIDWORKS SIMULATION are consistent with that obtained using with analytical calculation using MATLAB we can say that the simulation that we performed is correct and this verifies the results.