Analyzing a structure subjected to nonlinear, transient, dynamic, and impact loading is a majorleague task.
"We typically get requests for such work after clients have exhausted their internal sources," says President of Applied Analysis & Technology, David Dearth. But it doesn't have to be complex. "There is a way to sort of sneak up on solutions to nonlinear-transient-dynamic-impact problems using FEA techniques in a one-step-atatime approach." Dearth acknowledges that FEA idealizations of impacts can involve extended learning curves. "The biggest complaint I get from managers is that impact problems take too long to solve," he says. This installment presents one step. Others will follow.
The object of impact analysis is to predict peak deflections and stresses during the transient phase, when an object is vibrating or reacting to an impact. "A recent transient-impact problem involved a poppet valve, a hydraulic device that prevents fluid backflow. A particular design failed catastrophically after some period. It cracked and eventually broke a thin section when it slammed shut," he says.
To prepare for solving this problem, consider a long uniform bar traveling at a constant velocity that stops suddenly. "Estimate the peak translation or deflection of the free end of the bar during one complete cycle of the tension wave before the rod bounces off the rigid stop," says Dearth. He suggests neglecting damping. "Our solution will be valid for one complete wave reflection. In the real world, after the strain wave makes one complete reflection, the rod will bounce off the rigid stop. Dearth says this problem contains all the features of any real-life problem, and can be found in engineering literature and texts addressing impact.
To gain confidence in the solution, he presents two approaches: a manual solution using conventional equations, and a finite-element idealization.
Material properties for the problem are assumed constant and stress-strain relationships are linear. "In reality, geometries aren't simple and might need nonlinear-material effectsto account for localized yielding," he adds.
"Addressing the effects of viscous damping in the exact solutions by solving the differential equations of motion would make the problem an order of magnitude more complex. But to include damping in the FEA model, simply set the damping properties at about 5 to 7%."
Differential equations for predicting transient responses for the model are in the reference from Timoshenko. Dearth's calculation puts the theoretical exact solution for the rod tip's peak displacement (elongation) at 4.93288 in.
If you can't do a sample problem....
Dearth recalls comments from several analysts who insisted they "couldn't waste their time doing sample problems." Project managers must realize that their schedules have to reflect additional time and budget for the engineering staff to become proficient, comfortable, and confident in their analytical solutions. "My view is that if you can't develop accurate computer models of simple problems with known solutions, then you can't be confident doing real problems. And sample problems are the first step to solving impact problems, or any FEA problem for that matter. Test cases should be investigated prior to tackling real problems," he says.