Statics - Trusses

Truss Structures

Introduction

Fig. 1 Truss Structure in Nasville Airport

Fig. 2 San Jose Foot Bridge

Fig. 3 Comparsion of truss member and non-truss member

Trusses are one of the most common structures used in large scale construction but yet, they are actually one of the simplest to analyze due to a couple basic assumptions.

All joints are pinned
All members are two-force members
All loading is applied at the joints

These assumptions are simplifications of most structures, as can be seen in Fig. 1 and Fig. 2 where it is obvious that the joints are rigid. Furthermore, most loading is distributed along the length of the member, as opposed to just at the joints. Luckily, these assumptions are on the conservative side and can be assumed for basic analysis.

For all truss member, the load in each member can only be applied at the two ends of each member. If a third load is applied to the member as shown in the figure, then it can not be a truss member. With only two forces on the member, they must be collinear to be in static equilibrium.

Main concepts to remember

Forces only act at the pin joints (Fig. 3).
Member forces are always in the direction of the member.
There are no moments in a member (there can be moments on the full truss or a section of the truss if it has more than one member).

There are two methods to actually solve for each member load, the method of joints and the method of sections. These two methods are explained below.

Method of Joints

The method of joints examines each joint as an independent static structure. The summation of all forces acting on the joint must equate to zero. Both member forces and external forces are applied to the joint and then the force equilibrium equations are applied. For two dimensions, the equations are

SF_x = SF_y = 0

Since each joint must be in equilibrium these two equations can be applied to each joint. Generally, the support reactions should be determined first, and then the reaction joints should be solved. The joint sequence for solving can be important for multiple joints.

Likewise, for three dimensions each joint must also be in equilibrium. For 3D problems, however, the vector form is generally easier. The equation becomes

SF = 0 (three dimensional vector form)

Method of Sections

The method of sections splits the truss up into two or more parts and then applies both force and moment equilibrium equations. Because two-or three- dimensional sections are used instead of a one-dimensional point as with the method of joints, additional moment equations can be used to solve for truss member loads.

The biggest benefit of this method is the ability to solve for member forces in the middle of the truss without solving for a large number of joints.

First, the truss is split into two sections and all cut members are replaced with an unknown member load. Make sure that only three members are cut for a two-dimensional truss. The unknown member force are then treated as external loads and can be determined. Remember to solve for the support loads first since they will be needed in most problems to determine the cut member loads. The basic equations for each section are

SF_x = SF_y = 0 (two dimensional scalar form)

SM_z = 0 (two dimensional scalar form)

Zero Force Members

Fig. 4 Zero force Members

To help solve complex trusses, you should note which member loads must be zero (zero force members, Fig. 4) and eliminate them from the problem. A member that joins two other collinear members will be a zero force member. Also, two non-collinear members will be zero force members. All of these rules assume that there are no loads applied at the joint.

Influence Lines for Trusses

Fig. 5 A bridge

Fig. 6 The influence line of one of bridge members

In order to describe the force variation of a structure subject to a live or moving load, a influence line concept is used. An influence line represents the variation of either the reaction, shear, moment, or deflection at a specific point in a member as a point force moves over the member. From the influence line, it is easy to know how the moving load creates the influence for each member.

Trusses are assumed to support only axial loads. Hence, it is easy to construct the influence line for each member. For example, when a moving load passes the bridge (Fig. 5), the influence line of one of its members is Fig 6.

Truss Structures

		Introduction
Fig. 1 Truss Structure in Nasville Airport Fig. 2 San Jose Foot Bridge Fig. 3 Comparsion of truss member and non-truss member		Trusses are one of the most common structures used in large scale construction but yet, they are actually one of the simplest to analyze due to a couple basic assumptions. All joints are pinned All members are two-force members All loading is applied at the joints These assumptions are simplifications of most structures, as can be seen in Fig. 1 and Fig. 2 where it is obvious that the joints are rigid. Furthermore, most loading is distributed along the length of the member, as opposed to just at the joints. Luckily, these assumptions are on the conservative side and can be assumed for basic analysis. For all truss member, the load in each member can only be applied at the two ends of each member. If a third load is applied to the member as shown in the figure, then it can not be a truss member. With only two forces on the member, they must be collinear to be in static equilibrium. Main concepts to remember Forces only act at the pin joints (Fig. 3). Member forces are always in the direction of the member. There are no moments in a member (there can be moments on the full truss or a section of the truss if it has more than one member). There are two methods to actually solve for each member load, the method of joints and the method of sections. These two methods are explained below.

		Method of Joints
		The method of joints examines each joint as an independent static structure. The summation of all forces acting on the joint must equate to zero. Both member forces and external forces are applied to the joint and then the force equilibrium equations are applied. For two dimensions, the equations are SF_x = SF_y = 0 Since each joint must be in equilibrium these two equations can be applied to each joint. Generally, the support reactions should be determined first, and then the reaction joints should be solved. The joint sequence for solving can be important for multiple joints. Likewise, for three dimensions each joint must also be in equilibrium. For 3D problems, however, the vector form is generally easier. The equation becomes SF = 0 (three dimensional vector form)

		Method of Sections
		The method of sections splits the truss up into two or more parts and then applies both force and moment equilibrium equations. Because two-or three- dimensional sections are used instead of a one-dimensional point as with the method of joints, additional moment equations can be used to solve for truss member loads. The biggest benefit of this method is the ability to solve for member forces in the middle of the truss without solving for a large number of joints. First, the truss is split into two sections and all cut members are replaced with an unknown member load. Make sure that only three members are cut for a two-dimensional truss. The unknown member force are then treated as external loads and can be determined. Remember to solve for the support loads first since they will be needed in most problems to determine the cut member loads. The basic equations for each section are SF_x = SF_y = 0 (two dimensional scalar form) SM_z = 0 (two dimensional scalar form)

		Zero Force Members
Fig. 4 Zero force Members		To help solve complex trusses, you should note which member loads must be zero (zero force members, Fig. 4) and eliminate them from the problem. A member that joins two other collinear members will be a zero force member. Also, two non-collinear members will be zero force members. All of these rules assume that there are no loads applied at the joint.

		Influence Lines for Trusses
Fig. 5 A bridge Fig. 6 The influence line of one of bridge members		In order to describe the force variation of a structure subject to a live or moving load, a influence line concept is used. An influence line represents the variation of either the reaction, shear, moment, or deflection at a specific point in a member as a point force moves over the member. From the influence line, it is easy to know how the moving load creates the influence for each member. Trusses are assumed to support only axial loads. Hence, it is easy to construct the influence line for each member. For example, when a moving load passes the bridge (Fig. 5), the influence line of one of its members is Fig 6.