# Surface contact 101: Block and ramp with friction

July 7, 2011
To gain more confidence with FEA, it’s a good idea to first solve simple problems with pencil and paper.
 Authored by:David R. Dearth PresidentApplied Analysis & Technology Huntington Beach, Calif.Resources:Applied Analysis & Technology Edited by Leslie Gordon, [email protected]

Many less-experienced users of finite-element-analysis (FEA) software often have no clue if the results they get are anywhere near the correct answer. To gain more confidence with FEA, it’s a good idea to first solve simple problems with pencil and paper. Then, set up the problem in FEA and see if you get the same answer with both approaches.

As an example, let’s first manually solve the simple problem of how much horizontal force it takes to keep a block from sliding down a ramp. Most college students and even some high-school students encounter this problem while taking a first course in physics or engineering statics. The problem can be readily solved using simple free-body diagrams and summation of forces in two directions. But developing the solution using FEA can be tricky. How do you create an FEA idealization to perform a nonlinear static analysis using surface contact?

In the block-and-ramp problem, a block on a ramp has its weight acting down. Reaction forces comprise the normal force to the sliding surface of the block and the frictional forces acting parallel with the sliding surface. Frictional forces are defined as the normal force, N, multiplied by the coefficient of friction, μ, between the mating surface of the block and the ramp. The challenge is to determine the magnitude of the horizontal force, P, necessary to prevent the block from sliding down the ramp.

The solution comes from simple equations of statics:

1.) Sum the forces normal to the sliding surface, ΣFn = 0,

N – Psin30° – 500cos30° = 0

N = (P/2) + 433

2.) Sum the forces parallel to the sliding surface, ΣFp = 0, with μ= 0.30,

μN + Pcos30° – 500sin30° = 0

3.) Substitute for N: 0.3[(P/2) + 433]+0.866P-250 = 0

4.) Solve for P: P = 118.2 lb

These simple hand equations let users find the magnitude of the horizontal force, P, in a few minutes.

Solving this problem using FEA requires additional considerations to ensure the nonlinear solution will converge and prevent instabilities or singularities in the model. Therefore, rigid stops were added above and below the block to prevent excessive slippage should the FEA user enter too much or too little horizontal force to exert on the block. Again, the surface contact between the block and ramp was assumed to have a coefficient of friction µ = 0.30.

We developed an FEA idealization in Nastran to perform a nonlinear static analysis using surface contact. The block and the ramp are meshed using 2D solid plane strain elements. The rigid stops above and below the block use nonlinear gap elements with an initial gap set at 0.50 in. The rigid stops prevent the block from slipping only when the gap is completely closed. Lastly, the surface contact between the block and ramp uses nonlinear surface contact elements with the coefficient of friction between the mating surfaces again being µ = 0.30.

The user then types in an estimate for the magnitude of the horizontal force, P, and the software iterates to a solution. When P is too low, the block slides down the ramp until it hits the lower rigid stop.

In contrast, when P is too high, it pushes the block up the ramp until the block hits the upper stop.

Needless to say, when the user enters the correct value of P, the block is in equilibrium and remains in a stationary condition held by the horizontal force, P, and the frictional forces, μN. In this case, the software iterated to the solution 118.2 lb, which matched the results from solving the statics equations.

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