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# Methodology of structural analysis of load bearing systems

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The following information is intended to aid in the design of any static, non-moving load bearing system.


Knowledge and a basic understanding of Newton's laws, vector mechanics, calculus, and linear algebra is helpful in the study of


## Introduction

Load bearing systems exist all around us. Examples include: table, chair, wall, road, bridge, etc. They are typically designed to prevent failure: that is, the system should be strong enough to support the weight, or load, of that which it is supporting. A table is designed to withstand a certain amount of pressure (force/area); this upper limit of pressure is known as the maximum stress. If a load applied to the system exceeds this maximum stress, the system may fail (break). For example, a small computer table designed to hold at most 300 pounds will fail (collapse) if a 500 pound concrete block is placed on it. Loads can be forces, moments, or stresses acting on a system.

This type of analysis is useful especially in design for developing communities, where the cost of overestimation and conservative designs may not be affordable. A mechanical and structural analysis allows for the calculation of unknown quantities such as:

• The amount of cement and type of rebar needed to make a latrine slab
• The dimensions of resources to construct a house
• The minimum design requirements to sustain a load in order to potentially reduce costs of materials.

The first step to designing a load bearing structure is to understand all applicable loads the system will experience. The system and applicable forces can be modeled, simply, using a free-body diagram, or FBD.

Normal Force A force acting perpendicular to the cross section of the member on which it is acting (tensile = pulling, compressive = "pushing")
Sheer Force A force acting parallel to the cross section of the member on which it is acting
Moment Rotation caused by a force whose vector is perpendicular to axis of the member on which it is acting
Torsion Twisting about a member's axis due to a force; a torque is a moment whose vector is collinear with the axis of the member on which it is acting
Normal Stress Pressure caused by a normal force (tensile = pulling, compressive = "pushing")
Sheer Stress Pressure caused by a shear force or torsion

## Property of Materials

Modulus of elasticity (also known as Young's Modulus), modulus of rigidity, Poisson's ratio, tensile strength, and yield strength are a few material properties that are directly relevant to the scope of engineering mechanics. These material properties influence analysis and design and may be the reason one material is chosen over the other. For example, the modulus of elasticity of steel is three times that of aluminum, meaning it is more resilient to deformation due to loading than aluminum.

## Statics

External, Reaction, and Internal Forces and Moments [1] External forces and moments are the loads applied to a body. If the body is exposed to elements in the environment, it may experience wind loading. Reaction forces and moments are the forces exerted by supports. Internal forces and moments are inherent to a body and describe the body's resistance to applied loads. Internal forces can be modeled by making imaginary 'cuts' on a body or along a member in an FBD.

A system in static equilibrium is at rest. The net (sum of the) forces and net moments acting on a body in static equilibrium is zero.

$\Sigma\bar{F}=0$
$\Sigma\bar{M}=0$

Support & Connections Different types of connections are used to support certain loads. Structural supports make connections between structural members that allow, support, and prevent specific movements and loads. Common types of supports include: pin, roller, and fixed. Each type of support exerts a reaction (force). Pinned supports exert a vertical and a horizontal reactive force, roller supports exert a vertical reactive force, and fixed supports exert a vertical and horizontal force, as well as a moment. Supports are chosen based on the type of loading desired to be prevented.

Trusses Most familiar structure studied in statics is the truss. Trusses are supporting structures made of two-force members. Trusses make up larger structures such as bridges, buildings, transmission towers, etc. There are two methods of solving for the internal forces of a truss: the method of joints and the method of section.

Distributed & Concentrated Loads Loads can be concentrated at a point on an element or they can be distributed across a length or area. Distributed loads can be modeled as their statically equivalent load. For loads distributed across a length, the resulting force is the force multiplied by the distance on which it acts. When the distributed load is represented with vectors or a graph, the resulting equivalent force can be calculated by integrating the graph for the distributed force, or taking the area under the curve of the distributed force; this is especially helpful in the case the distributed load is non-uniform, varying across its length. The resultant equivalent force for loads that are distributed across an area is equal to the force multiplied by the area on which the force acts, or the volume under the curve that represents the distributed force.

The resulting equivalent force acts on the centroid of the distribution, which may be different from the centroid of the surface on which the force acts.

### Beams in Bending

Shear Force & Moment Diagrams

### Solar Photovoltaic Racking System

Typically, solar panels of an array sit on some sort of rail and are connected at discrete points either by means of a bracket, clip, or bolt through the frame. The load of the PV array is a uniformly distributed load (assumptions: all panels of equal weight and size, distributed uniformly about the plane centroid) across the plane on which it is installed, however, it can be modeled as point loads at the point of connections on each rail, when divided by the number of total connections.

Let's consider a ground mounted solar racking system, such as this one designed and created by solar racking company IronRidge.

### Failure

Failure Theories for Brittle & Ductile Materials
Factor of Safety
Stress Concentration

## References

1. Vable,M. (2002). Mechanics of Materials. New York, Ny: Oxford University Press