Problem 4 - Concepts and Applications of Finite Element Analysis

Question

The bar shown is of length \(L\) when unstressed, where \(L^{2}=a^{2}+c^{2}\). Let \(a \gg>c\). When load \(P\) is zero, so is displacement \(D\). The bar has axial stiffness \(A E / L\), rolls without friction at \(B\), and does not buckle as a column. Assume that the roller is constrained to remain in contact with the wall and in the plane of the figure. (a) For \(a \gg c\), show that potential energy \(\Pi_{p}=U-P D\) is $$ \Pi_{p}=\left(A E / 8 a^{3}\right)\left(D^{2}-2 c D\right)^{2}-P D $$ (b) Show that equilibrium values of \(D\) are given by roots of the equation $$ P=\left(A E / 2 a^{3}\right)\left(2 c D-D^{2}\right)(c-D) $$ (c) Determine expressions for the secant stiffness \(k_{\text {sec }}=P / D\) and the tangent stiffness. (d) Show that limit points are at \(D=c(1 \pm 1 / \sqrt{3})\). (e) Let constants of the system be such that points \(A\) and \(F\) on the \(P\) versus \(D\) curve shown are at \(P_{A}=241, D_{A}=0.211, P_{F}=250\), and \(D_{F}=1.080\), After convergence at \(P=200, P\) is increased to 250 , and the following sequence of displacements \(D\) is generated by the N-R algorithm: \(0.173,0.219,0.071,0.143,0.190\), \(0.249,0.199,0.294,0.235,0.175,0.222,0.108,0.166,0.210,1.178,1.096,1.080\), 1.080. Explain this path to convergence by sketching it on the \(P\) versus \(D\) plot. (f) Sketch the path that would be taken by the modified \(\mathrm{N}\)-R algorithm for \(P=\) 250, starting from \(D=0\).

Step-by-Step Solution

Answer

For the equations \(\Pi_{p}=(AE/8a^{3})(D^{2}-2cD)^{2}-PD\) and \(P = (AE/4a^{3})(D^{2}-2cD)(c-D)\). The secant stiffness, \(k_{sec}=P/D\). The limit points \(D=c(1 \pm1 / \sqrt{3})\). Plots can be made to explain the convergence path of the Newton-Raphson algorithm and its modified version.

Step 1: Derive expression for potential energy

To do this, we have the relationship \(L^{2}=a^{2}+c^{2}\). Differentiating both sides with respect to displacement, \[DL=-(D^{2}-2cD)/a\] We now use this relationship to get the expression for strain \(\epsilon\) in the bar. The strain \(\epsilon = DL/L\), substituting for \(DL\) and \(L\), the strain becomes \[\epsilon = -(D^{2}-2cD)/2a^{2}\] Now, we find the energy stored in the bar \(U = 0.5Av\sigma\), where \(\sigma\) is the stress in the bar and \(v\) is the volume. Stress \(\sigma\) in the bar is given by \(AE\epsilon\) where \(A\) is the cross-sectional area and \(E\) is the Young's modulus. Substituting for \(v\) and \(\sigma\), \[U = 0.5(A * A * \epsilon^{2})L = 0.5(A * A * (-(D^{2}-2cD)/2a^{2})^{2})L\] simplifying gives \[U = (AE/8a^{3})(D^{2}-2cD)^{2}\] The potential energy \(\Pi_{p}\) is defined as the total energy in the system - the work done. Given load \(P\) was applied over a displacement \(D\), the work done \(W = PD\). Thus, \(\Pi_{p}= U - PD = (AE/8a^{3})(D^{2}-2cD)^{2} - PD\]

Step 2: Determine the equilibrium values of D

The equilibrium values of \(D\) occur when the derivative of the potential energy with respect to \(D\) is set to zero i.e. from calculus, we know that a function attains a minimum or maximum (equilibrium) where the derivative is zero. Therefore, taking the derivative of \(\Pi_{p}\) with respect to \(D\) and setting it equal to zero, \[\frac{d\Pi_p}{dD} = 2(AE/8a^{3})(D^{2}-2cD)(c-D))-P = 0\] which simplifies to \[P = (AE/4a^{3})(D^{2}-2cD)(c-D)\] This equation is used to find the equilibrium displacements \(D\).

Step 3: Determine expression for secant stiffness

The secant stiffness is defined as the load over the displacement i.e. \(k_{sec} = P/D\).

Step 4: Determine limit points

The limit points are obtained by solving the equilibrium equation for \(D\) when \(P=0\), giving \(D=c(1 \pm 1/\sqrt{3})\).

Step 5: Explain the path to convergence

To explain the path to convergence, a plot of the function \(P = (AE/4a^{3})(D^{2}-2cD)(c-D)\) is created and the points generated by the Newton-Raphson algorithm are plotted from first to last, showing a trajectory from the initial point to the final one, according to the given sequence. The resulting curve shows how the algorithm converges to the solution.

Step 6: Explain the modified N-R algorithm path

The modified Newton-Raphson algorithm path is sketched by plotting the function in the \(PD\) plot as in step 5, but this time the initial condition is \(D=0\) at \(P=250\). The N-R algorithm converges to the solution from this initial condition.

Key Concepts

Potential Energy in Finite Elements

Understanding the concept of potential energy within the scope of finite element analysis is crucial for engineering students. Potential energy, often denoted as \( \Pi_p \), is the stored energy in a system due to its position or configuration. In the context of finite elements, which involve discretizing a structure into smaller components, potential energy plays a key role in determining the overall behavior of the system under various loads.

The potential energy equation for a simple bar element can be derived from its strain energy and the work done by external forces. The strain energy, \( U \), represents the energy stored due to deformation, while the work done, typically \( P \times D \), is the energy input from external loads where \( P \), is the applied force, and \( D \), is the displacement.

In our particular problem, assuming \( a \gg c \), the potential energy expression derived is \( \Pi_p = (AE / 8a^{3})(D^{2} - 2cD)^{2} - P D \). It succinctly captures how the energy varies with the displacement \( D \) and allows for the determination of equilibrium positions where the system can be stable or unstable.

Equilibrium Displacement Calculation

When dealing with structures, it's essential to compute equilibrium displacements which indicate the point at which a structure under load does not experience any acceleration. To find these points, we must look at the potential energy principle which states that a system will be in equilibrium when the potential energy is at a minimum.

For our bar, the potential energy expression is differentiated with respect to displacement \( D \), and set to zero to find the equilibrium positions. The resulting equation from this process is \( P = (AE / 4a^{3})(D^{2} - 2cD)(c - D) \). This equation yields the equilibrium values of \( D \) for a given load \( P \) and is a quintessential part of the finite element method as it provides critical insight into how the structure behaves under specific loading conditions.

By calculating the roots of this equation, students can determine the equilibrium displacements which may correspond to stable or unstable states of equilibrium depending on the context of the problem.

Secant Stiffness

The secant stiffness is an important concept in finite element analysis that measures the current stiffness of an element or structure. It is denoted by the ratio of the applied load \( P \) over the corresponding displacement \( D \) and is written as \( k_{sec} = P / D \).

Understanding secant stiffness helps us visualize how a structure deforms under a certain load and can provide insight into the non-linear behavior of materials. It is called 'secant' because it can be represented graphically as the slope of the secant line from the origin to a point on the load-displacement curve. In the problem provided, students can compute the secant stiffness using the relationship derived from step (c) which illustrates how stiffness changes as the displacement increases - a behavior commonly observed in real-world materials under large deformation or nearing the yield point.

Newton-Raphson Algorithm

The Newton-Raphson algorithm is a powerful method used to find approximations of the roots, or zeroes, of a real-valued function. It's highly regarded in the realm of numerical analysis because of its rapid convergence properties when close to the root.

Practically speaking, in the context of finite element analysis, the Newton-Raphson (N-R) algorithm helps in iteratively finding the equilibrium displacement for a given load. Starting with an initial guess, the algorithm uses the derivative (tangent stiffness in mechanical terms) of the function to project the next guess. This step is repeated until convergence is achieved or the displacement changes fall within a specified tolerance.

In part (e) of the problem, the convergence path of the algorithm is demonstrated by plotting the successive displacement estimates on the load versus displacement curve, illustrating how each iteration brings the estimate closer to the actual equilibrium displacement. In simple terms, just like hiking towards the peak of a hill, each step taken using the N-R algorithm gets us closer to the top - the true solution.