Problem 35 - Concepts and Applications of Finite Element Analysis

Question

The uniform cantilever beam shown carries a concentrated lateral force \(P .\) Model the beam by a single beam element. (a) Calculate \(w_{2}\) (the lateral deflection at the right end). Use the consistent load vector at node 2. Express your answer in terms of \(P, L ; E, I\), and \(x\). (b) Again calculate \(w_{2}\), but use load lumping: let the lateral force at node 2 be \(P x / L\) and ignore the moment load at node 2 . (c) Compute the exact \(w_{2}\) according to elementary beam theory. (d) Compute the ratio of the finite element \(w_{2}\) to the exact \(w_{2}\). Do this for part (a) and for part (b). Plot these ratios versus \(x / L\). (e) Compute the bending moment at the left end, first as given by the consistent loads, then as given by lumped loads. Compute the ratio of each moment to the exact value, and plot the two ratios versus \(x\) for \(0<\) \(x

Step-by-Step Solution

Answer

The step-by-step solutions involve applying the principles of beam theory, finite element analysis, and calculus to establish the relationships between forces, moments and deflections on the beam. These are then used to calculate the percentage error/ratio of the computed values to the actual known values.

Step 1 Compute consistent load vector-deflection

In order to compute the consistent load vector at node 2, we will use the formula for lateral deflection of a cantilever beam under a point load. This is given by the formula \(w_{2} = \frac{PL^3}{3EI}\) where \(P\) is the applied force, \(L\) is the length of the beam and \(E\) and \(I\) are the modulus of elasticity and moment of inertia respectively.

Step 2 Compute lumped load vector-deflection

In this section, we estimate the lateral force at the node by lumping the load. The force is given by \(P' = Px/L\). Hence the deflection by using the lumped load method is given by \(w_{2}' = \frac{P'L^3}{3EI} = \frac{PL^2x}{3EI}\).

Step 3 Compute the ratio of the computed deflection to the actual deflection

In order to compare the calculated deflections with the actual deflection, we will calculate the ratios of the calculated deflections to the actual deflection. This is done by simply dividing the computed deflection by the actual deflection. For the consistent load case: \(\frac{w_{2}}{w_{2_{exact}}}=\frac{PL^3}{3EI}\) / \(w_{2_{exact}}\) and for the lumped load case: \(\frac{w_{2}'}{w_{2_{exact}}}=\frac{PL^2x}{3EI}\) / \(w_{2_{exact}}\)

Step 4 Compute the Bending Moment

We calculate the bending moment at the left end using the bending moment formula. The bending moment resulting from a point load on a beam is given by the formula \(M = PL\).

Step 5 Compute the ratio of the computed bending moment to the actual bending moment.

We next compute the ratio of the calculated bending moment to the actual (given) moment. This can be done using the formula \(\frac{M}{M_{exact}} = (PL) / M_{exact}\).

Key Concepts

Cantilever Beam Deflection

Understanding the deflection of a cantilever beam is crucial in structural analysis. A cantilever beam is fixed on one end and free on the other. When a concentrated lateral force, denoted by \( P \), is applied at the free end, it causes the beam to bend. The deflection \( w_{2} \) at the free end of the beam can be calculated using the formula for a uniform beam's lateral deflection:\[w_{2} = \frac{PL^3}{3EI}\]This formula takes into account the modulus of elasticity \( E \) and the moment of inertia \( I \), which are properties of the beam material and cross-section, respectively. These properties influence how much the beam will bend when a load is applied.
It's important to use consistent units when applying this formula to avoid errors in the calculation of deflection.

• \( P \): Point load applied to the beam
• \( L \): Length of the beam
• \( E \): Modulus of elasticity
• \( I \): Moment of inertia

Bending Moment Calculation

The bending moment in a cantilever beam plays a significant role in determining the beam's structural response to loading. The bending moment at any point along the beam shows how the internal forces resist bending. For a concentrated lateral force at the free end of a cantilever beam, the bending moment \( M \) at the fixed end is calculated by:\[M = PL\]This simple product of the force \( P \) and the beam's length \( L \) indicates that a greater force or a longer beam will result in a larger moment.
Bending moments can lead to stresses in the beam that may cause failure, so understanding these stresses is essential for design safety. To compare the calculated bending moment with the theoretical value, one can calculate the ratio of the two:\[\text{Ratio} = \frac{M}{M_{exact}}\]This ratio helps evaluate the accuracy of the finite element model used for calculation.

Load Lumping Method

The load lumping method simplifies the process of analyzing beam deflection. Instead of considering the entire beam, this method breaks down the load into a simpler form. Typically, the force at the node is given by:\[P' = \frac{Px}{L}\]Where \( x \) is a position along the beam, and the load is distributed according to this equation. Using this approach, the deflection \( w_{2}' \) from load lumping at the free end of the beam is expressed by:\[w_{2}' = \frac{PL^2x}{3EI}\]This method is especially useful in finite element analysis when evaluating complex structures becomes computationally intensive. By simplifying the load distribution, engineers can efficiently study large-scale models while maintaining accuracy to a reasonable degree.

• \( P' \): Lumped load at a node
• \( x \): Distance along the beam
• \( L \): Length of the beam