Hello and welcome back to the drawing board with me Elliot Griffin. Today I'm going to talk to you about stiffness and preload. Stiffness is a measure of a members resistance to deformation and we're going to focus mainly on axial stiffness but the other ones that you may come across are bending and torsional or stiffness. Pre loading is something that we use to mitigate deflection of our props when they're in the ground. So stiffness in its simplest form can be described like a spring and you may recognize this diagram from an experiment that you might have done in school. It's a simple spring with the weight W on the end and we learn from this that when you apply a weight or a force to a spring that it deflects by x-mount distance. K is described as the spring constant or the spring stiffness of this system. Then we later learn that K can be defined not just by experimentation well it's a product of the cross sectional area of a member multiplied by the young's modulus of the material that that members formed from and divded by the length of the member. This then gives you the stiffness of a member which can be used to find out how much it will compress or stretch under a given loading.
Now in hydraulic propping it's not simple as a single spring, Steve's talked in previous videos about the hydraulic column of fluid that's at the end of our props that we use to fine tune the length and this is more often than not a lot less stiff than the rest of the steel extensions in the prop. So we have two springs in series and to calculate the overall stiffness of this prop we have to take the reciprocals of both springs. Now when we apply this formula to here and rearrange to make the resultant stiffness the subject, we end up with 1 divided by all these elements. We've got the length of the hydraulic column the Young’s modulus of the hydraulic fluid which we've obtained from test data and the area of the ram bore and we've also got the length, Young's modulus and the area of the steel extensions that were using in that prop. This then gives you a resultant stiffness for a single prop, which can be used to calculate how much it is going to deflect under a load.
Now when you're talking about an excavation you're not just talking about a single prop stiffness you're usually talking about the system stiffness. So it's made up of multiple props in that excavation and to calculate the system stiffness of multiple props it's a product of the stiffness of the props themselves but also the angle of the props and the spacing between props and also the loading conditions sometimes. This will then give you the system stiffness of your excavation support and depending on if you are using a structural steel or a hydraulic support system then you'll get different loadings and also different deflections because hydraulic systems are a little bit less stiff than steel systems are so this is when we use pre-loading to mitigate the deflection that happens due to that lower stiffness.
Now, to work out pre-loading that we require we can rearrange the formula for young's modulus and we concentrate here on the hydraulic column because this is the bit that we've got control over. So, to calculate the axial deflection of a single prop we take the force that’s applied to that prop, the area of the ram bore in that prop the length of the pump out and the young's modulus of the shoring fluid, this will then give us the deflection of a single prop under load in in an excavation. We can do this for multiple props in the excavation and pre-load them to reduce the deflection everywhere. We then take this deflection and we divide the forces applied to the prop by it, we can then multiply this by the amount of deflection that we need to reduce. This gives us a force that will be the pre-stress load that we need to put into the prop to mitigate the deflection that's happening. We know that this is a linear relationship with pressure so we can put the pressure straight back in to our struts and stop the excavation from moving in too much.
I hope this has given you a good outlook on stiffness of hydraulic propping and pre loading. I hope you liked the video and subscribe to the channel and join us next time on back to the drawing board.
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