Structural Calculations


I've put together the following information to assist in the calculation the maximum load that can be held by structural members in a design. It draws from various areas of structural engineering and mechanics, but is by no means exhaustive. It is also worth noting that there are many different ways to perform the analysis done below, and what follows is simply the options that I have chosen. It also skips a lot of the mathematical theory, to provide just the results required to perform the calculations.

There are two fundamental ways to approach the analysis:
1) You can find the forces present and dimensions used, and calculate if the material is strong enough,
2) Or you can use the specified strength of the material and work backwards to calculate the maximum forces and dimensions that can be used.

The second method is probably preferable in a design sense, as it allows you to design around the material that you have. However it is a little more complicated, as it requires the use of variables in you algebraic equations, which can become large. The first method allows you to simply use the equations and plug in numbers to calculate a numerical answer. We will use the first method here; all information required to perform the second method is available in the formulae.

Forces

The first step is to determine the forces acting on the structure. It is easiest to analyse one section of the structure in isolation, and then combine sections to analyse the structure as a whole. Ideally you want to find the critical section which will break first (because it is weakest and/or has the greatest load on it) - if that is strong enough then the other sections should be fine as well (as long as you chose the correct section to begin with.) Choosing the critical section to analyse is a field in itself and outside the scope of this page, largely because it depends entirely on the shape and design of the structure. If you are unsure and want to be safe, you can analyse all the (types of) sections in the structure, and the critical one(s) will become apparent.

I was always taught to draw a Free Body Diagram (FBD) of the structure to be analysed, which includes only the forces acting directly on that section. (For the purposes of this analysis, self-weight and the Normal reaction thereof will be ignored, as it is considered negligible compared to the forces from the intended usage.)

A Free Body Diagram of a simply supported beam under a point load might look like the following two examples:


 


Bending Moment Diagrams

For those interested in the theory, the BMD is the integral of the Sheer Force Diagram (SFD). Conversely the SFD can be obtained by differentiating the BMD. The SFD can also be obtained analytically by starting at the left hand side of the beam, and considering the sheer forces to the left as you move right along to the right hand side of the beam. This can be used to confirm the BMD diagram is correct (although it cannot check the constant of integration)

From the FBD above, the Sheer Force and Bending Moment Diagrams can be obtained as follows:





By the way, the above uses the following sign convention for positive Sheer Force and Bending Moment:


Note that a simply supported beam as above has pivot-type supports at the end, which allow the beam to bend freely, and only provide a reaction to the vertical forces. In most designs the ends of the beam are fixed, which provides and counter-acting moment as follows (please note that the following diagrams use the opposite sign convention to those above):



Note that the SFD would still be the same for the point load, and the BMD would still be the integral of the SFD - however the constant of integration is different.

Centroidal Axis

To calculate the Area Moment of Inertia the centroidal axis must be found. Assuming a uniform material density, the centroid is equal to the centre of mass of the shape. There are many ways to find this; I use the weighted average of area.  Remember you only need to find the y-coordinate of the centroid, if you are calculating the bending moment about the x-axis.

Area Moment of Inertia

The Area Moment of Inertia (aka the Second Moment of Area) is defined about a certain axis; I is used to denote the Moment of Inertia, with a suffix to denote the axis about which I is calculated. We will calculate the Moment of Inertia about the x-axis, Ix.

The Moment of Inertia of many objects can be calculated by starting with the formula for a rectangle:




The Parallel Axis Theorem may also be required to calculate the Moment of Inertia for your shape:

where  A is the cross-sectional area
and I is the perpendicular distance between the centroidal axis and the parallel axis.

Yield Stress

The yield stress of a material is the maximum stress that the material can support before it breaks. The units are force per area (ie same as pressure) because the greater the area of material, the more force it can withstand.

Basic steel is generally 250 grade, which means that the nominal yield stress is 250 MPa. (The actual yield stress is usually a bit higher - see Blue Scope Steel).

For a beam subjected to bending about one axis only, the formula for the yield stress is as follows:


where M = bending moment,
Ix = Area Moment of Inertia about x-axis
and y = distance from centroidal x-axis


If your calculations above show that the stress in the material is less than 250 MPa, congratulations! Your design is strong enough!

If not, you may have to add some reinforcing members to your design. Alternatively you can use 350 grade steel, which has a nominal yield stress of 350 MPa.

Safety Factor

In prudent engineering we usually use a safety factor (greater than 1) to allow for the conditions for which we have designed to be slightly exceeded. However for EV applications where we have to design for a deceleration of 20g, I believe this already incorporates a huge safety factor: -20g acceleration corresponds to going at 100kph and coming to a complete stop in 0.14 seconds: the car chassis would never withstand the forces required to decelerate the entire car so quickly, and the human body would be pulverised after anything like that kind of force, even with seatbelts and airbags etc. Thus I do not believe an additional safety factor is required in EV calculations.

References

The graphics used above were obtained from the following websites: