plane trusses - structures€¦ · plane trusses definition and assumptions ... james clerk maxwell...

Structures I University of Michigan, Taubman College Slide 1 of 45

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


Panels

Joints• Upper: U1, U2, U3...

• Lower: L1, L2, L3...

Members• Chords

• Web

Nomenclature


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


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


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

Not a true trussFrame structureRidgid Joints, moment connectionsFlexure members

Vierendeel “Truss”

Vierendeel bridge at Grammene, Belgium

Photo by Karel Roose

Salk Institute, La Jolla. Architect: Louis KahnEngineer: Komendant and Dubin


Graphic MethodsJames Clerk Maxwell 1869M. Williot 1877Otto Mohr 1887Heinrich Müller-Breslau 1904

Method of Joints

Method of Sections

Computer ProgramsDr. Frame (2D)STAAD Pro (2D or 3D)West Point Bridge Designer

Analysis

James Clerk Maxwell


1. Solve reactions (all external forces)

2. Inspect for zero force members

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


T – joints

L – joints

Inspection of Zero Force Members


1. Solve the external reactions for the whole truss.

Method of Joints - example


2. T or L joints by inspection.

3. Cut FBD of joint

4. Show orthogonal components

5. Solve by F horz. and vert.



Continue with joints having only one unknown in either horizontal or vertical direction. Generally work starting at the reactions.



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



Continue moving across the truss, joint by joint. Choose joints that have only one unknown in each direction, horizontal or vertical.



Solve the joints with the most members last.

Check that all forces balance.



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



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

Diagonals up toward center are in compression


Qualitative Force

For spanning trusses with uniform loading: (tension=blue compression=red)

Top and bottom chords greatest at center when flat (at maximum curvature)

Diagonals greatest at ends (near reactions)


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


1. Solve the external reactions for the whole truss.

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

Example – by method of sections


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.











5. Make final qualitative check of solution.



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


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


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


James Clerk Maxwell 1869M. Williot 1877Otto Mohr 1887Luigi Cremona 1890Heinrich Müller-Breslau 1904

Development of Graphic Statics

James Clerk Maxwell

Otto Mohr


Draw the truss to scale• Solve for all external forces on the truss.

• Make a simple line drawing of your truss.

• Draw it to scale.

Graphic Statics - Step 1


Letter external spaces• Label each space outside the truss with a letter.

• Start at bottom center (by convention).

• Continue in a clockwise direction around the outside of the truss.



Number internal spaces• Number each space inside the truss.

• Number from left to right.



Draw the force diagram• Start in the external space (A) and

move clockwise to the next space (B).

• Note the direction and magnitude of the external force you cross (12k up).

• Draw this force to scale.

• The force starts with the tail (A) and continues to the head (B).



Draw the force diagram• Continue clockwise around the

outside of the truss.

• Draw to scale the next force (6k downward) which is crossed as you move to the next space (C).

• Draw the tail at the first letter (B) and the head at the second (C).

• Draw the force in its actual direction and to scale.





• Draw to scale the next force (12k downward) which is crossed as you move to the next space (D).

• Draw the tail at the first letter (C) and the head at the second (D).






• Draw to scale the next force (6k downward) which is crossed as you move to the next space (E).

• Draw the tail at the first letter (D) and the head at the second (E).


• The head of the last force should land on the tail of the first force (the external forces close to zero). If not, then there was some error in the forces (like the reactions may be wrong)



Draw the member forces• Start in the external space (A) and

cross to the first adjacent internal space (1).

• Draw the line you cross starting at the external letter (A).

• Now move to the next external space (B) adjacent to the same internal space (1).

• Draw a line with the same slope as the member crossed starting at the letter of the external space (B).

• These two lines cross at a point (1).

• Note that you needed 2 lines to define the location of the point (1).



Draw the member forces• Now find the next point (2).

• Choose two spaces that are adjacent to space this next space (2).

• Now move to the next external space (A) adjacent to this internal space (2).

• Draw a line with the same slope as the member crossed starting at the letter of the external space (A).

• Now observe the member which is between the next two spaces (1 & 2).

• Draw a line with the same slope as the member through the location of the point which labels the adjacent space (1).

• The scaled length of the line equals the force. If a line has zero length (like 1-2) then the force in the member is zero.



Draw the member forces• Now find the next point (3).

• Choose two spaces that are adjacent to space this next space (3) which have points on the graph. (C&2)

• Find the member between the first pair of spaces (C&3). Draw a line with the same slope passing through the known point (C).

• Draw a second line with the same slope as the member crossed by the second pair (2&3) passing through the known point of the pair (2).

• Again, where these lines cross the unknown point can be found (3)



Draw the member forces• Continue moving across the truss

until all points corresponding to the interior spaces are determined

• Each line represents the scaled magnitude and direction of the force within corresponding member.



Reading the forces• The lines in the force diagram are scaled

to the forces in the corresponding members

• The slope of each line in both diagrams is the same.

• The force diagram makes a closed figure.



The geometry and force diagrams are duals. Changing one will result in changes in the other and vice versa. Design a truss to achieve desired forces by modifying the force diagram, and then modifying the geometry diagram to match.

Graphic Statics - Design


plane trusses - structures€¦ · plane trusses definition and assumptions ... james clerk maxwell...

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