Structures: Difference between revisions
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'''Part of the [[ | '''Part of the [[Statics|Statics]] course offered by the ''[[Applied Mechanics|Division of Applied Mechanics]]'', ''[[Engineering|School of Engineering]]'' and the ''[[Portal:Engineering and Technology|Engineering and Technology Portal]]''''' | ||
==Lecture== | ==Lecture== | ||
Structural engineering relies heavily on the strengths of materials and their ability to withstand forces of tension | Structural engineering relies heavily on the strengths of materials and their ability to withstand forces of tension, compression & shear. When used in conjunction with each other, as in the case of a truss, individual load bearing members both share and transmit loads, enabling the structure to accomplish much more than any individual member could alone. | ||
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===Plane Trusses=== | ===Plane Trusses=== | ||
One way of distributing | One way of distributing the force across a large distance is by building a ''Plane Truss'', which takes advantage of the principle of Equilibrium to translate forces along a system of interconnecting members. | ||
[[Image:Warren truss.PNG|thumb|400px|left|A Warren Truss]][[Image:Pratt truss.PNG|thumb|400px|right|A Pratt Truss]] | [[Image:Warren truss.PNG|thumb|400px|left|A Warren Truss]][[Image:Pratt truss.PNG|thumb|400px|right|A Pratt Truss]] | ||
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A simplest truss is a triangle made of three points: '''A''', '''B''' and '''C''', and three members: '''AB''', '''BC''' and '''CA'''. The method of distributing forces amongst many members relies on the engineer's ability to place a ''tensile'' or ''compressive'' force at any particular location. To properly sum forces at a particular point, one must be able to sum forces into that point and also away from it. Thus, '''AB''' and '''AC''' are in compression while '''BC''' is in tension. | |||
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Therefore, the Method of Joints may also be applied to a truss joint to determine the force (compressive or tensile) in each member of the truss. A place where the forces and members meet may become the joint to be examined ('''A'''), and the principle of equilibrium is applied, thus solving the question of where the forces are transmitted in that truss around that joint. | Therefore, the Method of Joints may also be applied to a truss joint to determine the force (compressive or tensile) in each member of the truss. A place where the forces and members meet may become the joint to be examined ('''A'''), and the principle of equilibrium is applied, thus solving the question of where the forces are transmitted in that truss around that joint. | ||
'''Example:'''<BR> | '''Example:'''<BR> | ||
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<math> \vec R_2 = 14.1 N</math>. | <math> \vec R_2 = 14.1 N</math>. | ||
We may sum the forces around an arbitrary point '''A'''. There are no <math>\ x</math> components of the force vectors, so <math> \sum \vec F_x \ = 0 </math>. | We may sum the forces around an arbitrary point '''A'''. There are no <math>\ x</math> components of the force vectors, so <math> \sum \vec F_x \ = 0 </math>. | ||
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<BR>Or...<BR> | <BR>Or...<BR> | ||
<math> \vec F_{AC} = \vec F_{AB} * sin{30} = 7.97N * sin{30} = 3.99N </math> ''(in Tension)'' | <math> \vec F_{AC} = \vec F_{AB} * sin{30} = 7.97N * sin{30} = 3.99N </math> ''(in Tension)'' | ||
Each successive member may be analyzed for its internal forces, be they ''compressive'' or ''tensile'' in a similar manner. For a successful analysis of every member of the truss, it is imperative that every external force be included. Therefore it is advised that the engineer begin analysis by the method of joints at each joint where external forces have been resolved and work his or her way inward towards the center. | Each successive member may be analyzed for its internal forces, be they ''compressive'' or ''tensile'' in a similar manner. For a successful analysis of every member of the truss, it is imperative that every external force be included. Therefore it is advised that the engineer begin analysis by the method of joints at each joint where external forces have been resolved and work his or her way inward towards the center. | ||
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If every joint in a structure does not need to be analyzed, yet the internal forces of some members need to be resolved, a potential shortcut method is the method of sections. The method of sections involves cutting a truss into a smaller part, thus changing the internal forces in the members in question to external forces, which may easily be solved by the method of Equilibrium. | If every joint in a structure does not need to be analyzed, yet the internal forces of some members need to be resolved, a potential shortcut method is the method of sections. The method of sections involves cutting a truss into a smaller part, thus changing the internal forces in the members in question to external forces, which may easily be solved by the method of Equilibrium. | ||
[[Image: | [[Image:LoadedTrussSection.JPG|200px|thumb|left]] | ||
'''Example:'''<BR> | '''Example:'''<BR> | ||
If the truss shown here were to need analysis for the internal forces on one particular member, ''' | If the truss shown here were to need analysis for the internal forces on one particular member, '''EF''', the method of sections could be used. Here we use the previous values for <math>\vec F_1 = 3N</math> and <math>\vec R_2 = 14N</math>. Also assume that each member is <math>\ 5</math> meters long. | ||
== | '''Solution:'''<BR> | ||
We solve the external forces on the truss completely. Equilibrium dictates that <math> \sum \vec F = 0 </math> and <math> \sum \vec M = 0 </math>. | |||
Once again, we first sum the ''moments'' about a particular point. In this case, let's use '''I''': | |||
<math> \sum \vec M_I \ = -\vec F_1*L + \vec F_{EH}*cos{30} = 0</math> | |||
<BR>Or...<BR> | |||
<math> \sum \vec M_I \ = -3N*5m + \vec F_{EH}*cos{30} = 0</math> | |||
<math> \vec F_{EH} = 17.3 N</math> (in tension). | |||
===Space Trusses=== | |||
[[Image:SpaceTruss.png|thumb|300px|left|A Triangle Space Truss]] | |||
A Space Truss is a 3D plane truss. Made up of similar structural forms, the space truss is a much more realistic form of truss thatn the Plane Truss. Calculation and resolution of loads, external and internal forces is performed the same way, however, care must be made to ensure that all three dimensions are correctly examined and properly resolved. Once again <math> \sum \vec F = 0 </math> and <math> \sum \vec M = 0 </math>. | |||
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==Assignments== | ==Assignments== | ||
'''Activities:''' | '''Activities:''' | ||
* Create an [[activity]] | * Create an [[Structures/activity|activity]] | ||
'''Readings:''' | '''Readings:''' | ||
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'''Study guide:''' | '''Study guide:''' | ||
# Wikipedia article:[[ | # Wikipedia article:[[Wikipedia:Plane Truss|Plane Truss]] | ||
# Wikipedia article:[[ | # Wikipedia article:[[Wikipedia:Tension|Tension]] | ||
# Wikipedia article:[[ | # Wikipedia article:[[Wikipedia:Compression|Compression]] | ||
# Wikipedia article:[[ | # Wikipedia article:[[Wikipedia:Method of Joints|Method of Joints]] | ||
# Wikipedia article:[[ | # Wikipedia article:[[Wikipedia:Method of Sections|Method of Sections]] | ||
# Wikipedia article:[[Wikipedia:Space Truss|Space Truss]] | |||
[[Category:Statics]] | [[Category:Statics]] | ||
[[Category:Advanced Classical Mechanics]] | [[Category:Advanced Classical Mechanics]] | ||
[[Category:Applied | [[Category:Applied mechanics]] | ||
[[Category:Engineering]] | [[Category:Engineering]] | ||