Stiffness Matrix for 1D Bar Element

Deriving the Entries of the Stiffness Matrix for a 1D Bar

Step 1: Understanding the Bar Element

Consider a 1D bar element of length L, with cross-sectional area A and Young’s modulus E. The element has two nodes, i and j, with displacements u_i and u_j at these nodes.

The force in the bar due to elongation (or contraction) is given by Hooke's Law as:

F = (EA * ΔL) / L

where ΔL is the change in length of the bar due to displacements at its nodes. We’ll use this to relate the displacements to internal forces and derive the entries of the stiffness matrix.

Step 2: Strain and Displacement Relation

If we define the displacements at the two nodes as:

• u_i: Displacement of node i
• u_j: Displacement of node j

The total elongation ΔL of the bar is:

ΔL = u_j - u_i

The strain ε in the bar is the change in length per unit length:

ε = (u_j - u_i) / L

Step 3: Stress and Force in the Element

The stress σ in the bar due to this strain is:

σ = E * ε = E * ((u_j - u_i) / L)

The internal force F in the bar is the stress times the cross-sectional area:

F = A * σ = A * E * ((u_j - u_i) / L) = (AE * (u_j - u_i)) / L

This internal force acts along the length of the bar and has equal and opposite effects on the nodes at either end. So we can represent the force vector for the bar element as:

\[ \begin{bmatrix} F_i \\ F_j \end{bmatrix} = \dfrac{AE}{L} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} u_i \\ u_j \end{bmatrix} \]

Step 4: Deriving the Stiffness Matrix Entries

From the equation above, we see that the stiffness matrix K for the 1D bar element is:

\[ K = \dfrac{AE}{L} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \]

Interpretation of Matrix Entries

• K_ii = (AE / L): This term represents the stiffness contribution of node i to the force at node i itself due to its displacement u_i. It captures how much force is needed at node i to displace it by a unit distance.
• K_jj = (AE / L): This term represents the stiffness contribution of node j to the force at node j due to its own displacement u_j.
• K_ij = -(AE / L): This off-diagonal term represents the stiffness relationship between node i and node j. It indicates the force at node i due to a unit displacement at node j. The negative sign shows that if node j moves in the positive direction, node i will experience a force in the opposite direction (compression).
• K_ji = -(AE / L): Similarly, this term represents the force at node j due to a unit displacement at node i.

Thus, the stiffness matrix K fully captures the relationship between forces and displacements at each node of the bar element.