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(a) Give a brief definition of what makes a structural finite element problem nonlinear, stating the 3 main sources of nonlinearity typically encountered and why they give rise to nonlinearity. (4 marks) (b) A nonlinear spring has a stiffness,

`k(u)`

, which is a function of displacement described below.

`A`

load of

`8N`

is applied to this spring:

`k(u)=(10)/((u+1)^(2)),(N)/(m)m`

Explain the operation of the Newton-Raphson method for such problems, and use it to obtain an estimate of the displacement by carrying out three iterations. State the force imbalance after the third iteration. Note: you may assume that the internal force is given by

`F_(int )(u)=10(u)/(u+1)`

but you may not directly solve the problem with this equation. ( 8 marks) (c) Bolted joints can be analysed using a variety of linear and nonlinear methods of varying complexity. Briefly discuss any approaches you know of, outlining why you might use each approach and evaluating them in terms of complexity.