# Design Cookbook: Beam Bending Examples

## Introduction

One very common problem that students have in 2.007 is not making their arm or structures stiff enough. This is a problem because the arms and structures usually need to move or support things. A lack of stiffness is very common cause of machine unreliability.

Remember from 2.001 that the following factors need to be known to calculate the stiffness of something.

The Young's Modulus [E]:
This is a material property that measures the stress/strain.
The Cross-Sectional Inertia [I]:
This is determined by the cross sectional geometry of the arm.
This gives your equation to calculate the the deflection. Typical configurations and their equations are listed below.

## Young's Modulus

 Material Young Modulus E Shear Modulus G Poisson's Ratio v ksi GPa ksi GPa Aluminum (pure) 10,000 70 3,800 26 0.33 Aluminum alloys 6061-T6 10,000 70 3,800 26 0.33 7075-T6 10,400 72 3,900 27 0.33 Steel 29,000 190-210 11,300 75-80 0.27-0.30 Delrin 3.1 0.35

## Moments of Inertia

Cross Section

Inertia

This is an approximation of a simple truss, ingoring the cross members. Both upper and lower members have the same area (A).

## Bending Configurations

v = deflection in y direction

v' = dv/dx = slope of deflection curve

db = v(L) = deflection at right end of beam

Theta b = v'(L) = angle at right end of beam

0 < X < A

A < X < L

Substitute a for b and (L-x) for x, to calc v and v' for a < x < L

## Tools for Evaluating Deflection

There are some tools to help you calculation the amount of deflection in your structures. If you forget how to uses any of the matlab functions you can type help <function name> for a description of how to uses that function.

 bending.xls calculates the cross sectional inertia for circular and rectangular sections. It also calculates the amount of delfection at a given point for a smiply supported and cantilevered beam.

 recinert.m and cirinert.m calculate the cross sectional inertia for rectangular and circular cross sections respectively. all units must be consistent. RECINERT(A,B,C,D) and CIRINERT(A,B)
»help recinert

RECINERT(A,B,C,D) calculates the cross sectional moment of interia for
a rectangular cross section
A = outer width, B = outer height
C = inner width, and D = inner height

To evalute a filled rectangle set C and D = 0

»help cirinert

CIRINERT(A,B) calculates the cross sectional moment of interia for
a pipe cross section
A = outer diameter, B = inner diameter

To evalutate a filled circle set B = 0

 cantbeam.m shows the deflection of a cantilevered beam loaded from 1 or more points. CANTBEAM(P,a,L,E,I,incr)
»help cantbeam

[def] = CANTBEAM(P,a,L,E,I,incr) returns arrays with
the deflection [def].  [def] is also plotted.

P is the force applied to the beam in NEWTONS
a is the distance from the left end that the force is applied in METERS
L is the length of the beam in METERS
E is the young's modulus of the material in Pa
I is the cross sectional inertia in METER^4
icnr is the number of increments to sample along the beam

Multiple forces can be entered in P, however, a must be the same length
to give a position for each force.  P can be positive or negative.

all values of a may not be greater then L
E, I, a, incr, and L must be greater then 0

def is in meters (plotted in mm)

Below is a sample plot from cantbeam.m for the following data:

L = 0.10 m
I = 2.1333e-11 m^4   (4mm by 4mm shaft)
E = 200E+9 Pa  (steel)
incr = 100
P = [-400 600 -200] N
a = [0.03 0.06 0.09] m

On the deflection plot the red lines are at the point of force application and they are "pushing" the beam (e.i. a positive force will have its red line below the beam "pushing" up). The values of each force is displayed at the end of its force line.

The deflection is plotted in mm but the array returned for [def] is in meters!

 simpbeam.m shows the deflection of a simple beam supported at either end, loaded from 1 or more points. SIMPBEAM(P,a,L,E,I,incr)
»help simpbeam

[def] = SIMPBEAM(P,a,L,E,I,incr) returns an array with
the deflection [def].  [def] is also plotted.

ALL CALCULATIONS ARE FOR A SIMPLE BEAM SUPPORTED AT EITHER END

P is the force applied to the beam in NEWTONS
a is the distance from the left end that the force is applied in METERS
L is the length of the beam in METERS
E is the young's modulus of the material in Pa
I is the cross sectional inertia in METER^4
icnr is the number of increments to sample along the beam

Multiple forces can be entered in P, however, a must be the same length
to give a position for each force.  P can be positive or negative.

all values of a may not be greater then L
E, I, a, incr, and L must be greater then 0

def arrary is in meters (plotted in mm)

Below is a sample plot from simpbeam.m for the following data:

L = 0.10 m
I = 2.1333e-11 m^4   (4mm by 4mm shaft)
E = 200E+9 Pa  (steel)
incr = 100
P = [-100 100 -100] N
a = [0.02 0.05 0.08] m

On the deflection plot the red lines are at the point of force application and they are "pushing" the beam (e.i. a positive force will have its red line below the beam "pushing" up). The values of each force is displayed at the end of its force line.

The plot may look like the beam is bending a lot, but compare at the scale on the x and y axis. In this example the maximum deflection is 0.075 mm for a 10 cm long beam!