Last edited 16dec10 by jmetzge2@illinois.edu

# Structural Analysis: Concrete Beam Deflection

## Introduction

My program was done in Python and the animation used VPython. The program displays a concrete beam supported by two columns. The columns and beams are designed to scale and oriented as a simple span setup. The arrow is labeled and represents the applied load on the structure. As the animation is run, the applied load moves vertically to illustrate the loads on the structure being applied to the system. The program is run by selecting run module from the Python toolbar (or pressing f5). The applied load hits the bottom of its path then returns to its origin. The distance which the arrow moves is a scaled up value of the maximum deflection the beam was calculated to withstand.

Here is a picture of the beam setup and the applied load.

### Background Information

The topic of structural analysis is one that has been investigated and reinvented over the years at the University of Illinois. Professors such as Hardy Cross have innovated new methods with which to quantify and explain the performance of structures in real world applications. Structures that are used on an every day basis, as well as structures that have not been erected yet. In order to create an everexpanding world of safe, yet exciting and ground breaking structural wonders, proper application of the techniques developed at Illinois (and elsewhere) must be implemented correctly and consistently. My project aims at inspecting an important, yet specific portion of analysis involved in structural mathematics, deflection. Deflection can be defined as the movement or displacement of an element or structure due to an external or internal force. Any structure, which for simplicity can be defined as a freestanding object comprised of individual elements, must be addressed case by case to properly analyze its behavior. Many factors will contribute to the final properties of a structure including orientation, material composition, external and internal loads, connections, and mathematical assumptions.

In order to compose a simple and effective analysis of structure deflection, I have implemented a set of parameters that will govern the boundaries of my project. The first boundary addresses the setup of the structure in question. The deflection of a single beam will be analyzed, and this beam will be supported by two individual columns on each end of the beam. Both columns will be considered infinitely strong and able to support the beam under any applied load, this way, a determinate analysis of the horizontal member alone can be achieved. The beam will be simply supported by each of the columns along a given span width. Now that the setup has been defined, the material composition must be chosen. The mathematics that accompany structural analysis of concrete members present some interesting relationships, so for my project a concrete beam and columns will be used. The specific parameters of the concrete beam are what will contribute to its behavior under loading, including width, depth, steel reinforcing, length, and concrete strength. In my analysis I have defined these properties as constants in order to isolate the loading parameter as a variable. In depth exploration of these parameters and properties are included in the mathematical development section.

Once the boundaries and constants of the setup have been defined, several structural equations can be applied to figure the final deflection experienced by the beam. My project aims to keep everything but the external load constant in order to see the effect on the given beam. In order to help the user see the effect of a varying external load on beam deflection, I have used the VPython programming environment for visualation. A simple beam setup has been modeled with a moving arrow to show increasing force and the resulting deflection. As the force magnitude varies, so does the deflection experienced by the beam. The proportion between the applied force and the resulting beam deflection has been calculated in the next section, and subsequently applied to the programming used for the visualization. One important thing to note is that the beam deflections experienced by most concrete beams are so small that they cannot be detected by visual inspection. The programed visualization has been scaled up several times in order that the user may see the deflection resulting from an increasing load.

### Coding

The following is the code used to make the animation. It was a fairly simply while, if statement that moved the arrow a certain distance before reversing the velocity and moving it back to the original position.

from visual import * wallR = box(pos=(5,0,0), size=(1,10,1), color=color.white) wallR = box(pos=(-5,0,0), size=(1,10,1), color=color.white) mybox = box(pos=(0,5.5,0), length=11, height=1, width=1) pointer = arrow(pos=(0,12,0), axis=(0,-6,0), shaftwidth=1, color=color.yellow) mybox = box(pos=(0,-5,0), length=12, height=1, width=5, color=color.green) forcelabel = label(pos=pointer.pos, text='Applied Load', xoffset=3, yoffset=5, space=1, height=10, border=6, font='sans') pointer.velocity = vector(0,-3,0) deltat = 0.005 t = 0 while t < 2.5: rate(50) if pointer.pos.y < 8: pointer.velocity.y = -pointer.velocity.y if pointer.pos.y > 12: pointer.velocity.y = -pointer.velocity.y pointer.pos = pointer.pos + pointer.velocity*deltat t = t + deltat