Crankshaft Design - Stress Analysis
What is desired from a crankshaft, apart from the obvious function, is for it to be of sufficient strength to last theoretically an infinite number of cycles under normal operating conditions. The forces applied to the crankshaft, mainly, are alternating; compression to tension; gas force to inertia force. This cyclic loading is of course a typical case for fatigue failure.
Calculating the alternating stresses and comparing them to the allowable stress range of the material used (fatigue strength or endurance limit) using a Goodman diagram, allows a Fatigue Factor of Safety (FFoS) to be determined. This factor can be reduced to a minimum acceptable limit to ensure only the minimum amount of material is being used, therefore not making an unnecessarily heavy crankshaft. The value for the FFoS is down to the designer or company or manufacturer. Obviously it needs to be above 1 if it is to be considered capable of lasting an infinite number of cycles. A typical figure may be in the region of 1.3 to 1.8.
 "Point Loading of Crank")
Point Loading of Crank
So firstly, calculating the alternating stresses requires a number of calculations. The way the forces are applied to the crank is simplified into point loads, as illustrated, where the load is on the centre of the pin and there are reaction forces at the main journals either side. These forces will make the crank bend.
 "Critical Section")
Critical Section
The neutral axis is between the pin and main journal. To simplify this further, it is assumed that this is a simple beam bending problem, where the section of the beam is the critical section which is the minimum cross section between the point loads, which ends up being a section through the fillets as shown left.
Using simple beam bending formulae, the stress in the fillet can be calculated for both load cases:
σbending = My/I
where, σbending = bending stress [N/M2], M = bending moment [NM], y = distance from centroid to extreme fibre [M], and I = second moment of area of beam (critical) section [M4].
So far, all this could be done simply by hand on paper if the geometry of the crankshaft is assumed to be simple, i.e. the pin, main journal and web all being simple shapes, resulting in a rectangular critical section. The second moment of area for this simple rectangular section would be:
I = bd3 / 12
where, b and d are the width and thickness of the critical section.
However, as the design progresses, the critical section may no longer be a simple rectangular section, as shown in the figure 'Critical Section'. Having this crank now in 3D CAD will enable us to interrogate the geometry and get certain properties out of the critical section. This is of course dependant on the CAD you use; I was using Pro/Engineer which had the functionality to interrogate a cross section and provide useful information. One item that is very useful to us here is the section modulus, which is simply the ratio of I/y and it is represented by 'Z'.
Therefore we can rewrite the formula for stress as:
σbending = M/Z
 "Maximum Stress and Bending Moment")
Maximum Stress and Bending Moment
The bending moment is the force of the reaction at main journal multiplied by the distance from centroid of the critical section perpendicularly to centre of main journal (where reaction force is). This will now calculate stress in extreme fibre which is basically the skin of the fillet. Calculate this for both maximum net gas load and maximum inertia load.
Forces 'up' (inertia) put the fillet into tension and are therefore considered positive, forces 'down' (gas) are compressive and therefore considered negative.
To calculate the FFoS with these alternating forces, a few more things are required:
- - Mean stress = (max + min) / 2 (remembering max is positive tension and min is negative compression)
- - Alternating Stress = (max - min) / 2
- - Material Fatigue strength and UTS
Now we are almost ready to calculate the FFoS. The FFoS is calculated by:
FFoS = σalw / σalt
where, σalw = Allowable Stress Range; and σalt = Alternating (Actual) Stress Range.
The allowable stress range is basically the material.s fatigue limit or strength which has been altered to take the mean stress into account, as the material fatigue limit is assuming zero mean stress. To calculate allowable stress range:
σalw = Fatigue Strength (UTS - σmean / UTS)
To calculate the Alternating Stress Range:
σalt = [(σmax * Kf) * (σmin * Kf)] / 2
where, Kf = Altered Stress Concentration Factor.
The Stress Concentration Factor, Kt, effects the value of calculated stress and is a direct result of the geometry of an object. Refer to any engineering design handbook or the Bosch handbook to see many examples of stress concentration factors. "Peterson's Stress Concentration Factors" by Walter D. Pilkey is a fine book for this and has a section devoted to crankshafts.
The Notch Sensitivity, q, of a material determines how much the SCF effects the calculated stress. This effect will result in Kf.
Kf = 1 + q (Kt - 1)
One more thing to note, the UTS and Fatigue Strength of the material may also need to be adjusted to suit any treatments designed to increase the strength of the material. For example, heat treatment, nitriding, fillet rolling maybe used and should be taken into account. What the factors are for these treatments is down to consulting the person responsible for the treatment. As an example, the fillet rolling may have the effect of doubling the UTS and fatigue strength. It would only be locally in the fillet, but that is where we are calculAting the stress. So when calculating σalw, the UTS and Fatigue Strength should be altered by these factors.
From here, now we know the current FFoS, we can make a decision if the crankshaft meets the minimum requirements. If not, then we can go back and modify journal sizes, materials or geometry and see the effects on the FFoS and find a solution.
