Deformation of a Beam
? 2000 Harald Kammerer, GERB Schwingungsisolierungen, Germany http://www.gerb.com
Introduction
This worksheet shows the calculation of the deformation of a simply supported beam
The following Maple techniques are highlighted:
The Problem
Consider an elastic both side simply supported beam. This beam can be for example a bridge. It is loaded by a moving load. This might be a car driving over the bridge. The situation is considered to be quasi static. So there are no dynamic effects respected.
NOTE: Influence of shearing is not considered.
The following variables are used:
F: Force
xF: Position of the force related to the left end of the beam
L: Length of the beam
x: Position where the deflection of the beam is calculated
EI: Product of the elastic modulus and the moment of inertia
Further variables are defined at there first appearance.
All units are m and N
Set Parameters of the Problem
Parameter
Value
Length of the beam
Force
Number of steps for the load position
Length of steps between load positions
=
Modulus of elasticity
Dimension of the Cross Section
The Cross section is for later calculations divided in three parts, the plate and the two webs.
The dimensions of the parts of the cross section are collected in sets.
Boundaries of the cross section in z-direction
Collect the coordinates centers of gravity of every part in sets (arbitrary with respect to the left upper corner, other points are also possible)
Procedures
All the main calculations and graphics are programmed in the form of procedures.
To execute the code , left-click the icon.
To view the code, right-click the icon, then select "View Code Edit Region"
Procedures for the Calculation
Area of the Cross Section
Calculating the cross sectional area of a beam made up of three rectangular pieces.
#crosssec
Center of Gravity
Calculating the center of gravity.
#cog
Moment of Inertia of a Beam
Calculating the moment of inertia of the beam.
#inertia
Bending Moment of a Beam
Simply supported beam at both ends with a single point load acting on it.
#moment
Simply supported beam at both ends with a single point load acting on it (shear deformations are neglectable in this example).
#deform
Normal Stress of a Beam
#stress
Procedures for the Plots
Rotation and Following Translation of the Coordinate of a Point
This procedure is used for the later graphical representation of the solution
#transrot
Drawing a Car in a x-y-Coordinate System
This procedure is used to draw a car which symbolizes the moving load in the later graphical representation of the solution.
#carplot
Draw the Supports of the Beam
#support
Draw the Bridge Head on Both Sides
#bridgeplot
Plot the Deformation
In this procedure the deflection of the beam will be plotted. To show the current position of the load there are a little car an an arrow drawn.
#plotsol
Plot the Normal Stress in the Beam
In this procedure the normal forces in the beam will be plotted. To show the current position of the load there is an arrow drawn.
#stressplot
Execution
Calculation
Calculate the center of gravity by using the procedure cog.
Boundaries of the cross section in z-direction in relation to the center of gravity
Calculate the moment of inertia by using the procedure inertia.
In this example only the moment of inertia with respect to the horizontal axis is necessary.
Area of the Cross section of the beam
For the further calculation the product of elastic modulus and moment of inertia is interesting.
Calculate the bending moment depending on x and xF.
Calculate the deformation of the beam depending on x and xF
Separate the deflection
Separate the gradient
Calculate the normal stress in the beam depending on x and xF.
Create the Animation
Calculate the line of deflection for every position of the load and return the solution in form of a plot by using the procedure plotsol. Thereby the load is moving over the bridge like a driving car. The deflection of the bridge is 50 000 times inflated. Note: there are no real dynamic effects regarded. When the car reaches the right end of the bridge, it drives back to the left end. The result is shown by an little animation.
#line of deflection
Calculate the normal stress for every position of the load and return the solution in form of a plot by using the procedure stressplot.
#normal stress
View the Result
To view the animations, left-click the plot and press in the animation toolbar at the top of the document.
Deformation
The maximum deflection of the beam is reached when the load is positioned in the middle of the beam. The maximum deflected point is then the center of the beam. The maximum deflection is thus
Normal Stress
The stress for the situation when the load is positioned in the middle of the beam at the center of the beam is
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