Here is an example of using the DefaultThrowableRenderer class from the logback-classic library in Java:

import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import ch.qos.logback.classic.spi.IThrowableProxy;
import ch.qos.logback.classic.spi.StackTraceElementProxy;
import ch.qos.logback.classic.util.DefaultThrowableRenderer;

public class LogbackExample {
    public static void main(String[] args) {
        Logger logger = LoggerFactory.getLogger(LogbackExample.class);
        try {
            throw new Exception("This is an example exception");
        } catch (Exception e) {
            IThrowableProxy throwableProxy = new ThrowableProxy(e);
            DefaultThrowableRenderer defaultRenderer = new DefaultThrowableRenderer();
            StackTraceElementProxy[] stepArray = throwableProxy.getStackTraceElementProxyArray();
            String[] renderedStackTraces = defaultRenderer.render(throwableProxy);
            for (String renderedStackTrace : renderedStackTraces) {
                System.out.println(renderedStackTrace);
            }
        }
    }
}

This example demonstrates how to use DefaultThrowableRenderer class to render the stack trace of an exception in to a String array, by calling the render(IThrowableProxy) method, passing in an IThrowableProxy instance. IThrowableProxy is an internal class of logback-classic, representing a proxy for a throwable object,

In this example, an instance of the DefaultThrowableRenderer class is created and then the exception is caught and its stack trace is passed to the render() method. The method returns the stack trace as an array of strings, which are then printed to the console.

You can use the output generated by this class to include stack trace of the exception in your custom log appender or layout.

Please note that this is a basic example, In real-world scenarios, you may want to customize the way the stack traces are rendered or pass additional options to the renderer.