I need assistance calculating the bending stress in a wheel axle. The load is applied through two bearings mounted on a sleeve. Presumably, the load is uniformly distributed through the sleeve along the axle. Although the sleeve continues up to the mounting plates on either side, in actuality some bending exists. How is the bending stress calculated? Specifically, how do I determine the distance (or gap) to the bending load? I've heard a rule of thumb is to use 1/2 or 3/4 the shaft diameter as an offset distance. Do I measure the distance to the bending load from the inside edge of the side plates or from the center of the plates?
Generally rules of thumb are dependent on some limited range of geometries, materials, environment, etc. So it is difficult to say if the rule you mentioned applies. If the drawing you showed is an accurate representation of the geometry the shear effects will not be insignificant. A Finite Element solution would give you an answer, but unless you understand what is happening fundamentally you won't know if the answer is meaningful. Also, in an FEA bending stresses are often masked by other effects such as bearing stresses. I don't know you're background and experience, so I'll give you the fundamentals. The best way to visualize how a structure like this works is to draw a free body diagram. Start with the bearings on the sleeve and determine how the loads will get from the bearings to the sleeve, and from the sleeve to the axle. You will quickly see that how the loads distribute will depend greatly on radial gaps and fits, and on relative stiffnesses. What can be said is that, for bending stresses, it would be conservative to assume that the bending loads are located and distributed below the bearings only. A somewhat less conservative solution would be to assume a uniform load distribution below the bearings, and then tapering linearly to zero over some length (you could use the bearing width to taper it over or take it all the way to the ends of the sleeve - again, this length is dependent on relative stiffnesses). Ultimately, you will need to get a "feel" for your structure. If you are uncertain about how strong it is, make conservative assumptions about how it will react the loads.
First thing to do to figure out your bending stress is figure out your bending moment by drawing a shear diagram and bending moment diagram (see the two links below - the first gives a little overview on shear and moment diagrams, the second is an online calculator that will let you input your loads and supports and determine your resulting bending moments). http://en.wikipedia.org/wiki/Shear_and_moment_diagram http://bendingmomentdiagram.com A sketch or diagram of what your arrangement would help in making sure you have an accurate free body diagram to accomplish the items above.
You need a free body diagram of your wheel - but when you are drawing this free body diagram don't forget the tangential force and lateral force - as well as the radial load. These forces will all impose forces and moments and torques on the axle. The lateral force will come from cornering forces - and the tangential force will come from the braking force - or accelerating force - or steady state force - to overcome the "drag of the vehicle" (air drag and rolling resistance drag). You may also have to consider the dynamic forces from the wheel too - as these may not be insignificant (it will depend on the particular problem/application). The picture below - depicts the forces I have mentioned above. http://www.google.co.uk/imgres?imgurl=http://upload.wikimedia.org/wikipedia/en/thumb/6/65/Tire_Force_Variation1.jpg/494px-Tire_Force_Variation1.jpg&imgrefurl=http://en.wikipedia.org/wiki/Tire_uniformity&h=599&w=494&sz=33&tbnid=ytbrxLjEsmQqsM:&tbnh=90&tbnw=74&zoom=1&usg=__BIELRFF1NE8v-i85uutNT87OURw=&docid=r2nTacup3xNpuM&sa=X&ei=4eecUpWwO8-QhQel9IGwDA&ved=0CHUQ9QEwBw
Great diagram Lochnagar. To add to my previous post - there will be bending moments in more than one plane if your loading is like that in Lochnagar's drawing - so the bending stress you need to account for is the vector sum of the bending moments due to the lateral, tangential, and radial loads.