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An axial bar as in Figure 1 with span $b$ is made of material with modulus of elasticity $E$ and lineathy varied cross section area given by $A(x)=(1−x/L)A_{1}+(x/L)A_{2}=A_{1}+(Λ_{2}−A_{1})_{x}$ in which $A_{1}$ and $A_{2}$ are the area of lett and right end sections respectively. The bar is modeled 83 one bar element with the two ends as nodes. Construct the stiffness matrix of the element.
An axial bar as in Figure 1 with span $b$ is made of material with modulus of elasticity $E$ and lineathy varied cross section area given by $A(x)=(1−x/L)A_{1}+(x/L)A_{2}=A_{1}+(Λ_{2}−A_{1})_{x}$ in which $A_{1}$ and $A_{2}$ are the area of lett and right end sections respectively. The bar is modeled 83 one bar element with the two ends as nodes. Construct the stiffness matrix of the element.
Figure 1: Cantilever Beam Structure

To construct the stiffness matrix of the element, we first need to derive the element stiffness matrix for an axial bar with a constant cross-sectional area. We can then modify this matrix to account for the linearly varying cross-sectional area

The element stiffness matrix for an axial bar with the constant cross-sectional area is given by

Where

Follow these steps as shows