Calculating the dead load of a beam

[bluespark1978]

Hi there, electrician in need of some mechanical study help.

The question I have is,

A beam has a solid cross sectional area of 100mm and is simply supported by 2 supports 3m apart. Calculate the dead load that can be safely supported when applied to the middle of the beam.


The only calculations I can find in my notes require a value for the KN per m so I am at a complete loss and I'm afraid I don't even have any attempts to show you.

If anyone can help it would be very much appreciated.

Thanks

The Attempt at a Solution

 

[LawrenceC]

You will need more information such as the moment of inertia of the beam or the dimensions of its cross section so you can determine the moment of inertia. You have to know whether the beam is simply supported or cantilevered. You must know the material of the beam so you can determine the allowable stresses.

The area would not have a dimension of mm. It would be a unit of length squared.

 

[bluespark1978]

Thanks for your reply Lawrence, This course is a nightmare everything is so badly worded that i spend more time reading on the internet instead of the course notes. they have not supplied any details of what the beam is made from but I assume from the way it is worded that it is simply supported.

 

[LawrenceC]

So the beam could be either wood or a steel I beam. That is not much to go on and certainly not enough to do an analysis. Nevertheless, I'll outline how to determine the maximum stress in a simply supported beam with the weight of the supported mass in the center.

Since it's in the center, each support supports 1/2 the weight. Since it's simply supported, the supports do not exert a resisting moment so the moment diagram looks like an upside down V with the peak moment right under the load. The peak moment is w*L/2 where w is the weight of load and L is the distance between supports.

The formula for bending stress is M*c/I where M is the moment, c is the distance from the center of the beam (assuming symmetry here like an I beam) to the surface and I is the second moment of inertia. You have to know the dimensions of the cross section to determine the inertia or even look it up somewhere. Note that the units of Mc/I is force per unit area which is the unit of stress. Once you compute the stress, you go to a table and see what the yield stress is for the material you are using and apply whatever safety factor that is appropriate. The bending stress is tensile on the bottom of this beam and compressive on the top of this beam. In magnitude, they are equal.

 

[bluespark1978]

Hi Lawrence,

Thanks very much for all the info, feels a bit cleared in mind now, will see how i get on.

 

[bluespark1978]

Hi Lawrence

I have been having another go at this and just wondered if you think I am on the right track or not, here are my workings.

Information.

Cross section of beam = 100mm x 100mm
length of beam = 3m or 3000mm
material is mild structual steel
safety factor for dead load = 0.25
ultimate shear strength = 320Nmm^2 so working shear stress = 80Nmm^2
yield stress = 190Nmm^2 so working yield stress = 47.5Nmm^2
ultimate bending strength = 480Nmm^2 so working bending stress = 120Nmm^2
modulus of elasticity = 80000Nmm^2

What I know i can calculate

2nd moment of area ( I )

I = D^4 / 12

I = 100^4 / 12

I = 8333333mm^4

1st moment of area ( y )

y = 0.5 x D

y = 0.5 x 100mm

y = 50mm

Section modulus ( Z )

Z = I / y

Z = 8333333mm4 / 50mm

Z = 166667mm^3

so taking that w = the force of the beam ( unsure of which value to use so calculated them all )

w = f x A

Using working stress for bending.

w = 120Nmm^2 x 100mm^2 = 1200000Nmm

using working stress for shear

w = 80Nmm^2 x 100mm^2 = 800000Nmm

using working stress for yield

w = 47.5Nmm^2 x 100mm^2 = 475000Nmm

The bending moment will then be M = wl^2 / 8.

 

[LawrenceC]

You have some incorrect units. Yield stresses, etc, should be in the units of force/d^2 or Newtons/mm^2. It's force per area, not force times area.

"The bending moment will then be M = wl^2 / 8"

Here, your units are force-length squared. The unit of bending moment is force-length.
You need to explain what w is. You have it as force times area.

The importance of correct units cannot be overemphasized when it comes to computation.

The first thing you need to do is compute the maximum moment on the beam. It will occur at the center. Compute that first based on a load applied at the center. Then compute the bending stress based on that maximum moment. The upper surface of the beam will be in maximum compression while the lower surface will be in maximum tension. Also compute the shear stress.