It’s often the case in the metal workshop that the laws of physics are challenged in order to pursue a design aesthetic. Luckily, with some knowledge of physics and a little common sense we can employ these laws and prevent a lot of wasted time and effort.
In the case below, a square section tubular steel construction is required to support the weight of a person. It’s quite clear that if a person stands on the extended tube (fig.a) the rest of the frame will tip up (fig.b)
We will need to support the tube under her weight to prevent this from happening (fig.c below)
I know what you’re thinking; that the frame could still tip up from the back and she could fall forwards. Yes; do you see how our basic understanding of the laws of physics, are deeply routed within us?
So we have now arrived at fig.d a design a long way from the original, but one that will prevent the frame from tipping and be able to support the load.
However, this is not acceptable from the designer’s perspective and only fig.a will do.
We will need to prevent the frame from tipping and this can be done in a couple of obvious ways; by either counterbalancing the frame (fig.e) or bolting the frame to the floor (fig.f) this being the most acceptable from the designers point of view and leads us to our next problem:
What is the maximum distance the unsupported steel tube can extend from the rest of the frame before it will bend under the weight of the person standing on it?
And I suppose now we come to the essence of the document, as this simple question is not easy to answer and one that requires a much greater appreciation of the laws of physics and a determination in employing them to our advantage. Luckily the internet is a rich source of information and in particular http://www.physicsforums.com who’s members BTown and xxChrisxx did the hard work in providing me with the information necessary to answer the question.
First we must understand turning forces as it is clear that this is what we are referring to. The extended tube (acting as a lever) can no longer turn the frame and tip it up as it is bolted to the floor. However, when the woman stands on the tube the force to turn the frame is still there.
This turning force is called a ‘moment’ and is measured in Newtons per metre squared or Nm2. ‘Moments’ are calculated by multiplying the length of the lever (here the extended tube) by the amount or force applied to it (represented by the weight of the woman). However, weight is not the same as force. Weight is static, it has magnitude (measured in Kilograms) but it is going nowhere, whereas force (measured in newtons) has both magnitude and direction.
Fortunately, Isaac Newton can help us here: he states that 1kg falling to Earth has a force of 9.81N
Because the woman is raised up, she has a downwards force acting on the tube. Lets say she weighs 70kg (the average weight of a person) she has a force of 70 X 9.81 = 686.7N
If she stands ½ a meter away from the fixed point she exerts 0.5 x 686.7 = 343.35N/m2 of turning force
This helps us find the moment of turning force along any given point on the extended tube she stands on, but it doesn’t help us find out if this moment of turning force exceeds that of which the tube is capable of withstanding. For this we require some information about the properties of the material
The Yield Strength
The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically (bend and not return). Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed.
“Yield (engineering).” Wikipedia
There is no need to calculate this, it should be provided as a specification by the supplier of the material.
The yield strength was found here: http://www.roymech.co.uk/Useful_Tables/Matter/Strength_st.htm
The value depends on the quality of the material. In this instance the value of the yield strength was rated at 280Mpa (mega pascals) that of a construction quality steel.
Second Moment of Inertia
Also known as the second moment area, the area moment of inertia, moment of inertia of plane area, or second area moment, is a property of a cross section that can be used to predict the resistance of beams to bending and deflection, around an axis that lies in the cross-sectional plane.
“Second moment of area.” Wikipedia
There’s no need to calculate this either as once again it should be provided as a specification by the supplier of the material (see below).
Electro Resistance Welded (ERW) Square Hollow Section Table of Dimension and Properties
Area of Section
Second Moment of Area
mm x mm
The table above (an excerpt of http://www.roymech.co.uk/Useful_Tables/Sections/SHS_hf.html) refers to the steel tube that will be used in the final construction. The properties will change if using Rolled Hollow Section (RHS) even though the dimensions might stay the same, so it’s important to make sure you have the correct table of data.
There are 3 columns that are of interest to us:
- Column 1 the size of the tube
- Column 2 The thickness of the tube wall (T)
- Column 3 Second Moment of Area
You can see in bold the row which we will be referring too as we will be using 25mm x 25mm with a 2mm wall thickness.
We also need to know the distance from the centre of the tube (x-x) to the outside edge in this case 12.5mm (fig.g) below.
The next step is to refer to the following equation and do some math. However, this is where my knowledge falters and I would really appreciate some input from someone who knows more about this subject than I do.
In the equation below * has been used as a multiplication symbol rather than x so as not to confuse it with x which is the unknown value we are trying to find.
Yield strength of steel = (moment of turning force)*(half the material thickness ÷ second moment of inertia)
We don’t know the amount (or ‘moment’) of turning force required to meet that of the yield strength of the steel as this is what we are trying to find out. But we do know that it is Force multiplied by an unknown distance. So we keep these two things apart in the equation as shown below in (696.7N/m2 * x)
Yield strength of this steel = 280Mpa
Force = 686.7N/m2
Distance = x (This is what we are trying to find out)
Half the material thickness =12.5mm
Second moment of inertia =1.56cm4
So the equation reads: 280Mpa = (686.7N/m2 *x)(12.5mm÷1.56cm4)
In the equation above you will notice (12.5mm ÷ 1.56cm4) we cant divide millimetres by centimetres so we will have to convert 1.56cm4 to mm which is 156000mm
280Mpa = (686.7N/m2 *x)(12.5mm÷156000)
280*15600 = 12.5*686.7*x
3244800 ÷ (12.5*686.7) = x
3244800 ÷ 8583.75 = x
378.01 = x
Or x (the yield point at which the metal will bend and not recover) = 378.01mm
As you can see, I’ve done the math without really explaing what has happened. This is because I do not know and am merely repeating what was explained to me on the physics forum by xxChrisxx and BTown. If anyone wishes to elaborate please feel free.