In the weighted residuals approach, the displacement field in the trial solution is expressed as a sum of nodal displacements multiplied by shape functions. Which of the following expresses this form?

In the weighted residuals approach, the displacement field in the trial solution is expressed as a sum of nodal displacements multiplied by shape functions. Which of the following expresses this form?

• A. u = Σ N_a u_a
• B. v = Σ w_b δ u_b
• C. u = Σ N_b u_b
• D. u = Σ N_a δ u_a

Answer: (A)

In the weighted residuals approach, the displacement field is approximated by a linear combination of shape functions, each associated with a node, multiplied by the nodal displacement at that node. This builds the interior field from the unknown nodal values, ensuring the field matches the nodal data exactly at the nodes. So the trial displacement is written as u_h(x) = ∑_a N_a(x) u_a. The other forms mix different fields or notation (such as using a different variable or a delta increment), which do not represent the trial displacement field.

In the weighted residuals approach, the displacement field is approximated by a linear combination of shape functions, each associated with a node, multiplied by the nodal displacement at that node. This builds the interior field from the unknown nodal values, ensuring the field matches the nodal data exactly at the nodes. So the trial displacement is written as u_h(x) = ∑_a N_a(x) u_a. The other forms mix different fields or notation (such as using a different variable or a delta increment), which do not represent the trial displacement field.