plane trusses method of joints
TRANSCRIPT
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Structures I University of Michigan, Taubman College Slide 1 of 19
Architecture 314
Structures I
Plane Trusses
Method of Joints
Definition and Assumptions
Nomenclature
Stability and Determinacy
Analysis by joints
Phaeodaria – Ernst Haeckel
2 Force Members
Pinned Joints
Concurrent Member Centroids at Joints
Joint Loaded
Straight Members
Small Deflections
Bullring Covering, Xàtiva, Spain
Kawaguchi and Engineers, 2007
Definitions and Assumtions
Structures I University of Michigan, Taubman College Slide 2 of 19
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Panels• Secments: left to right
Joints• Upper: U1, U2, U3...
• Lower: L1, L2, L3...
Members• Chords
• Web
Nomenclature
Structures I University of Michigan, Taubman College Slide 3 of 19
2D Trusses• Concurrent Coplanar
3D Trusses• Concurrent Non-Coplanar
Force Systems
University of Michigan Architectural Research LabUnistrut System, Charles W. Attwood
Foster Bridge, 1889Ann Arbor, Michigan
Structures I University of Michigan, Taubman College Slide 4 of 19
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For:
• j joints
• m members
• r reactions (restraints)
Three conditions
• m < k unstable
• m = k stable and determinate
• m > k stable and indeterminate
Stability and Determinacy
Structures I University of Michigan, Taubman College Slide 5 of 19
Quiz
For each of the following trusses, determine whether they are:
A) Stable
B) Unstable
• m < k unstable
• m = k stable and determinate
• m > k stable and indeterminate
Truss 1
Truss 2
Structures I University of Michigan, Taubman College Slide 6/42
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Not a true trussMoment frame structureRigid joints as moment connectionsFlexure in members
Vierendeel “Truss”
Vierendeel bridge at Grammene, Belgium
Photo by Karel Roose
Salk Institute, La Jolla. Architect: Louis KahnEngineer: Komendant and Dubin
Structures I University of Michigan, Taubman College Slide 7 of 19
Method of Joints
Method of Sections
Graphic MethodsJames Clerk Maxwell 1869M. Williot 1877Otto Mohr 1887Heinrich Müller-Breslau 1904
Computer ProgramsDr. Frame (2D)STAAD Pro (2D or 3D)West Point Bridge Designer
Analysis
James Clerk Maxwell
Structures I University of Michigan, Taubman College Slide 8 of 19
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1. Solve reactions (all external forces)
2. Inspect for zero force members (T’s & L’s)
3. Cut FBD of one joint
4. Show forces as orthogonal components
5. Solve with FH and FV (no M)
6. Find resultant member forces (Pythagorean Formula)
Method of Joints – procedure
Structures I University of Michigan, Taubman College Slide 9 of 19
T – joints
L – joints
Inspection of Zero Force Members
Structures I University of Michigan, Taubman College Slide 10 of 19
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1. Solve the external reactions for the whole truss.
Method of Joints - example
Structures I University of Michigan, Taubman College Slide 11 of 19
2. T or L joints by inspection.
3. Cut FBD of joint
4. Show orthogonal components
5. Solve by F horz. and vert.
Method of Joints - example
Structures I University of Michigan, Taubman College Slide 12 of 19
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Continue with joints having only one unknown in either horizontal or vertical direction. Generally work starting at the reactions.
Method of Joints - example
Structures I University of Michigan, Taubman College Slide 13 of 19
Continue moving across the truss, joint by joint. Solve by FH and FV .
Method of Joints - example
Structures I University of Michigan, Taubman College Slide 14 of 19
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Continue moving across the truss, joint by joint. Choose joints that have only one unknown in each direction, horizontal or vertical.
Method of Joints - example
Structures I University of Michigan, Taubman College Slide 15 of 19
Solve the joints with the most members last.
Check that all forces balance.
Method of Joints - example
Structures I University of Michigan, Taubman College Slide 16 of 19
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Inspect the final solution to see that it seems to make sense.
Method of Joints - example
Structures I University of Michigan, Taubman College Slide 17 of 19
Qualitative T or C
For typical gravity loading:(tension=red compression=blue)
Top chords are in compression
Bottom chords are in tension
Diagonals down toward center are in tension (usually)
Diagonals up toward center are in compression (usually)
Structures I University of Michigan, Taubman College Slide 18 of 19
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Qualitative Force
For spanning trusses with uniform loading: (tension=blue compression=red)
Top and bottom chords greatest at center when flat (at maximum curvature or moment)
Diagonals greatest at ends (near reactions, i.e. greatest shear)
Structures I University of Michigan, Taubman College Slide 19 of 19