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Architecture 314

Structures I

Plane Trusses

Definition and Assumptions

Nomenclature

Stability and Determinacy

Analysis by joints

Analysis by sections

Analysis and design by graphics

2 Force Members

Pinned Joints

Concurrent Member Centroids

Joint Loaded

Straight Members

Small Deflections

Bullring Covering, Xàtiva, Spain

Kawaguchi and Engineers, 2007

Definitions and Assumtions

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Panels

Joints• Upper: U1, U2, U3...

• Lower: L1, L2, L3...

Members• Chords

• Web

Nomenclature

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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

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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

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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

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Not a true trussMoment frame structureRidgid 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

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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

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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

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T – joints

L – joints

Inspection of Zero Force Members

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1. Solve the external reactions for the whole truss.

Method of Joints - example

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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

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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

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Continue moving across the truss, joint by joint. Solve by FH and FV .

Method of Joints - example

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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

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Solve the joints with the most members last.

Check that all forces balance.

Method of Joints - example

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Inspect the final solution to see that it seems to make sense.

Method of Joints - example

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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)

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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)

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1. Solve Reactions

2. Cut section through member

3. Choose point where all but one of the unknown forces cross and M

4. Continue with FH and FV

Method of Sections - procedure

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1. Solve the external reactions for the whole truss.

Sum moments about each end. Or using symmetry, divide vertical forces evenly between reactions

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Method of Sections - example

2. Solution proceeds by cutting FBDs of either joints or sections of the truss.

Member forces are shown as horizontal and vertical force components at each cut section.

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Method of Sections - example

75k 75k

2. Solution proceeds by cutting FBDs of either joints or sections of the truss.

Member forces are shown as horizontal and vertical force components at each cut section.

3. Choose a point where all but one of the forces cross and sum moments.

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Method of Sections - example

75k 75k

83.9k

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4. Continue with FH and FV

Member forces are shown as horizontal and vertical force components at each cut section.

Method of Sections - example

75k 75k

83.9k

80.3k

37.5k

4. Continue with FH and FV

Member forces are shown as horizontal and vertical force components at each cut section.

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Method of Sections - example

75k 75k

83.9k

80.3k

37.5k

70.3k

37.9k

2. Solution proceeds by cutting FBDs of either joints or sections of the truss.

Member forces are shown as horizontal and vertical force components at each cut section.

3. Choose a point where all but one of the forces cross and sum moments.

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Method of Sections - example

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4. Continue with FH and FV

Member forces are shown as horizontal and vertical force components at each cut section.

Method of Sections - example

5. Make final qualitative check of solution.

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Method of Sections - example

Howe Truss

1. Cut a panel with diagonals

M at L2 and resolve upper chord force at U2. This gives U1U2H

M at U1 to find L1L2

M at U2 and resolve U1L2 at L2 to find U1L2H

M at L0 and resolve U1L2 at L2 to find U1L2V

6. U1U2V can now be found by FV

Tips on Sections

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Parker Truss

1. Cut a panel with diagonals and M at L2 to solve U1U2H as before.

M at U1 to find L1L2

M at U2 and resolve U1L2 at L2 to find U1L2H

4. Find point x in line with U1U2. M at x and resolve U1L2 at L2 to find U1L2V

5. U1U2V can now befound by FV

Tips on Sections

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K Truss

1. Make cut A-A to avoid the mid panel joint

M at U1 to get L1L2

M at L1 to get U1U2

4. The vertical web forces can be solved using joints

5. Cut B-B through the diagonals

M at U2 and resolve lower diagonal

at L2 to find its H component. The V

component can be found by slope

triangle. Top and bottom chords are

known from steps 2. & 3.

7. Repeat step 6 by M at L2 to find

other diagonal.

Tips on Sections

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