https://r2.erweima.ai/mathbot/image/1724826290243ptmvrwp5.pdf

please solve each question using R studio

The uploaded file contains a PDF of an exam for a Numerical Methods course for Civil Engineers. The document includes two main questions, each with several parts. Let's solve each question using R Studio, as requested.

### **Question 1: Bending Moment Diagram and Trapezoidal Rule**

#### **1.1 Line Plot of the Bending Moment Diagram**

The bending moment $M(x)$ for a simply supported beam under a uniformly distributed load is given by the equation:

\[
M(x) = 4M_{\text{max}} \left(\frac{x}{L}\right)\left(1 - \frac{x}{L}\right)
\]

where:
- $L = 10$ m (beam span)
- $M_{\text{max}} = 100$ kNm (maximum bending moment at the center)
- $n = 10$ (number of divisions for the trapezoidal rule)

To create the line plot, we need to calculate the bending moment at each point along the length of the beam and then plot these values.

**R Code to Create the Plot:**

```r
# Define parameters
L <- 10 # Length of the beam in meters
Mmax <- 100 # Maximum bending moment in kNm
n <- 10 # Number of divisions

# Define the x values (positions along the beam length)
x <- seq(0, L, length.out = n + 1)

# Calculate the bending moment M(x) at each x
M_x <- 4 * Mmax * (x / L) * (1 - x / L)

# Create the plot
plot(x, M_x, type = "b", pch = 16, col = "blue",
     xlab = "Length (m)", ylab = "Bending Moment (kNm)",
     main = "Bending of a Simply Supported Beam")
```

#### **1.2 Calculate the Area Using the Trapezoidal Rule and the Exact Area**

To calculate the area under the curve $M(x)$, we use the trapezoidal rule:

\[
A \approx \frac{h}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)
\]

where:
- $h = \frac{L}{n}$ is the width of each trapezoid
- $f(x) = M(x)$

The exact area can be found by integrating the function $M(x)$ over the interval $[0, L]$.

**R Code to Calculate Area and Percentage Error:**

```r
# Define the width of each trapezoid
h <- L / n

# Calculate area using the trapezoidal rule
A_trap <- h * (0.5 * M_x[1] + sum(M_x[2:n]) + 0.5 * M_x[n + 1])

# Calculate exact area using integration
A_exact <- integrate(function(x) 4 * Mmax * (x / L) * (1 - x / L), 0, L)$value

# Calculate percentage error
percentage_error <- abs((A_exact - A_trap) / A_exact) * 100

cat("Area using Trapezoidal Rule:", A_trap, "\n")
cat("Exact Area:", A_exact, "\n")
cat("Percentage Error:", percentage_error, "%\n")
```

#### **1.3 Change $n$ to 5 and Calculate the New Percentage Error**

Repeat the steps in 1.2 but with $n = 5$.

**R Code:**

```r
# Redefine n and recalculate x and M_x
n <- 5
x <- seq(0, L, length.out = n + 1)
M_x <- 4 * Mmax * (x / L) * (1 - x / L)

# Recalculate width of each trapezoid
h <- L / n

# Recalculate area using the trapezoidal rule
A_trap <- h * (0.5 * M_x[1] + sum(M_x[2:n]) + 0.5 * M_x[n + 1])

# Recalculate percentage error
percentage_error <- abs((A_exact - A_trap) / A_exact) * 100

cat("Area using Trapezoidal Rule with n = 5:", A_trap, "\n")
cat("Percentage Error with n = 5:", percentage_error, "%\n")
```

### **Question 2: Solving the ODE for the Water Table Height**

Given the second-order ODE for the steady-state height of the water table:

\[
K \bar{h} \frac{d^2h}{dx^2} + N = 0
\]

with boundary conditions $h(0) = 10$ m and $h(1000) = 5$ m, and parameters:
- $K = 1$ m/d
- $N = 0.0001$ m/d
- Average height $\bar{h} = \frac{h(0) + h(1000)}{2} = \frac{10 + 5}{2} = 7.5$ m

**R Code to Solve the ODE and Plot the Results:**

```r
# Load necessary library
library(deSolve)

# Define the ODE system
model <- function(x, state, params) {
  h <- state[1]
  y <- state[2]
  K <- params[1]
  N <- params[2]
  h_bar <- params[3]
  
  dh <- y
  dy <- -N / (K * h_bar)
  
  list(c(dh, dy))
}

# Define parameters and initial conditions
params <- c(K = 1, N = 0.0001, h_bar = 7.5)
initial_state <- c(h = 10, y = 0.001666667)

# Define the sequence of x values
x <- seq(0, 1000, by = 1)

# Solve the ODE
solution <- ode(y = initial_state, times = x, func = model, parms = params)

# Extract the height of the water table (h) from the solution
h_values <- solution[, "h"]

# Plot the results
plot(x, h_values, type = "l", col = "red", lwd = 2,
     xlab = "Distance (m)", ylab = "Water Table Height (m)",
     main = "Water Table Height vs Distance")
```

These R scripts will perform the necessary calculations and produce the required plots for both questions.

Would you like further details or have any specific questions?

### 5 Related Questions:
1. What are the advantages of using numerical methods like the trapezoidal rule over analytical solutions?
2. How does the choice of $n$ in numerical integration affect the accuracy of the results?
3. What other numerical methods can be used for solving second-order ODEs?
4. How can numerical instability affect the solution of differential equations in R?
5. What are some real-world applications of bending moment diagrams in civil engineering?

**Tip:** Always validate your numerical results with analytical or known solutions to ensure accuracy and reliability.

Explore how to solve numerical methods problems using R Studio. Includes detailed solutions for bending moment diagrams and ODEs with examples of trapezoidal rule integration. Suitable for college-level students and engineers.

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