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Structural Analysis

Engineering Mechanics:

Statics in SI Units, 12e

Copyright 2010 Pearson Education South Asia Pte Ltd

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Chapter Objectives

Determine forces in members of a truss Learn about

the method of joints the method of sections

Analyze forces acting on the members of frames andmachines composed of pin-connected members

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Chapter Objectives

Determine forces in members of a truss

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Chapter Outline

1. Simple Trusses2. The Method of Joints3. Zero-Force Members4. The Method of Sections5. Space Trusses6. Frames and Machines

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6.1 Simple Trusses

A truss composed of slender members joined togetherat their end points

Planar Trusses

Planar trusses support roofs and bridges Roof load is transmitted to the truss at joints by means

of a series of purlins

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6.1 Simple Trusses

Planar Trusses

The analysis of the forces developed in the trussmembers is 2D

Similar to roof truss, the bridge truss loading is alsocoplanar

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6.1 Simple Trusses

Assumptions for Analysis

1. All loadings are applied at the joint- Weight of the members is neglected

2. All members are joined by smooth pins- Assume connections provided the center lines of the

joining members are concurrent

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6.1 Simple Trusses

Simple Truss

The truss must be rigid to prevent its collapse The simplest form that is rigid or stable is a triangle

roller support

pin joint

pin joint

roller support

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6.2 The Method of Joints

We need to know the force in each member: For applied forces, equations of equilibrium can be

applied

Forces in individual members are internal forces For each joint:

Fx = 0 and Fy = 0 must be satisfied for equilibrium

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Example 6.1

Determine the force in each member of the truss and

indicate whether the members are in tension or

compression.

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Example 6.1

Determine the force in each member of the truss and

indicate whether the members are in tension or

compression.

Ax

Ay Cy

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Solution

2 unknown member forces at joint B 1 unknown reaction force at joint C 2 unknown member forces and 2 unknown reaction

forces at point A

For Joint B,

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Solution

For Joint C,

For Joint A,

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Solution

For Joint C,

For Joint A,

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Solution

FBD of each pin shows the effect of all the connected membersand external forces applied to the pin

FBD of each member shows only the effect of the end pins on themember

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6.3 Zero-Force Members

Method of joints is simplified using zero-forcemembers

Zero-force members do not support any load In general, when 3 members form a truss joint, the 3rd

member is a zero-force member provided no externalforce or support reaction is applied to the joint

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Example 6.4

Using the method of joints, determine all the zero-force

members of the Fink roof truss.

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Example 6.4

Assume all joints are pin connected.Assign reaction forces

Ax

Ay Ey

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Solution

For Joint G,

Therefore, GC is a zero-force member.(note: each pin joint is in equilibrium)

For Joint D,

Therefore, DF is a zero-force member.

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Solution

For Joint F,

For Joint B,

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Solution

Fy = 0 applied to FHC will show that HC is not a zero-

force member.

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6.4 The Method of Sections

If a body is in equilibrium, then any part of the body is inequilibrium

To find forces within members, an imaginary section is used tocut each member into 2 and expose each internal force as

external

cut

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Consider the truss and section a-a as shown

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Procedure for Analysis

Equations of Equilibrium

Summed moments about a point

Find the 3

rd

unknown force from moment equation

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Example 6.5

Determine the force in members GE, GC, and BC of the

truss. Indicate whether the members are in tension or

compression.

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Solution

Choose section a-a since it cuts through the threemembers

Draw FBD of the entire truss

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Solution

Draw FBD for the section portion

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6.5 Space Trusses

Consists of members joined together at their ends toform 3D structure

The simplest space truss is a tetrahedron Additional members would be redundant in supporting

force P

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6.5 Space Trusses

Assumptions for Design

Members of a space truss is treated as 2 forcemembers provided the external loading is at the joints

When weight of the member is considered, apply it asa vertical force, half of its magnitude applied at eachend of the member

Method of Joints

Solve Fx = 0, Fy = 0, Fz = 0 at each joint Force analysis has at least 1 unknown force and 3

unknown forces

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6.5 Space Trusses

Method of Sections

When imaginary section is passes through a truss itmust satisfied

Fx = 0, Fy = 0, Fz = 0

Mx = 0, My = 0, Mz = 0

By proper selection, the unknown forces can bedetermined using a single equilibrium equation

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Example 6.8

Determine the forces acting in the members of the space

truss. Indicate whether the members are in tension or

compression.

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Solution

For Joint A,

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Solution

For Joint B,

To show,

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6.6 Frames and Machines

Composed of pin-connected multi-force members Frames are stationary Apply equations of equilibrium to each member to

determine the unknown forces

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6.6 Frames and Machines

Free-Body Diagram

Isolate each part by drawing its outlined shape show all forces and couple moments act on the part identify each known and unknown force and couple

moment

indicate any dimension apply equations of equilibrium assumed sense of

unknown force or moment

draw FBD

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Example 6.9

For the frame, draw the free-body diagram of (a) each

member, (b) the pin at B and (c) the two members

connected together.

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Solution

Part (a)

BA and BC are not two-force AB is subjected to the resultant forces from the pins

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Solution

Part (b)

Pin at B is subjected to two forces, force of the memberBC and AB on the pin

For equilibrium, forces and respective componentsmust be equal but opposite

Bx and By shown equal and opposite on members AB

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Solution

Part (c)

Bx and By are not shown as they form equal butopposite internal forces

Unknown force at A and C must act in the same sense Couple moment M is used to find reactions at A and C

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QUIZ

1. One of the assumptions used when analyzing a simple

truss is that the members are joined together by

__________.

A) Welding B) Bolting C) RivetingE)Smooth pins E) Super glue2. When using the method of joints, typically_________ equations of equilibrium are applied at every

joint.A) Two B) Three

C) Four D) Six

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QUIZ

3. Truss ABC is changed by decreasing its height from H

to 0.9 H. Width W and load P are kept the same. Which

one of the following statements is true for the revisedtruss as compared to the original truss?

A) Force in all its members have decreased.B) Force in all its members have increased.

C) Force in all its members have remained the same.

D) None of the above.

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QUIZ

4. For this truss, determine the number of zero-force

members.

A) 0 B) 1 C) 2D) 3 E) 4

FF

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QUIZ

5. Using this FBD, you find that FBC = 500 N. Member

BC must be in __________.

A)TensionB) Compression

C) Cannot be determined

6. For the same magnitude of force to be carried, truss

members in compression are generally made _______ as

compared to members in tension.

A) ThickerB) Thinner

C) The same size

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QUIZ

7. In the method of sections, generally a cut passesthrough no more than _____ members in which the forcesare unknown.

A) 1 B) 2

C) 3 D) 48. If a simple truss member carries a tensile force of Talong its length, then the internal force in the member is______ .

A) Tensile with magnitude of T/2

B) Compressive with magnitude of T/2C) Compressive with magnitude of T

D) Tensile with magnitude of T

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QUIZ

9. Can you determine the force in member ED by making

the cut at section a-a? Explain your answer.

A) No, there are 4 unknowns.

B) Yes, using MD = 0 .

C) Yes, using ME = 0 .

D) Yes, using MB = 0 .

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QUIZ

10. If you know FED, how will you determine FEB ?

A) By taking section b-b and using ME = 0

B) By taking section b-b, and using FX = 0 and FY = 0

C) By taking section a-a and using MB = 0

D) By taking section a-a and using MD = 0

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QUIZ

11. As shown, a cut is made through members GH, BG

and BC to determine the forces in them. Which section

will you choose for analysis and why?

A) Right, fewer calculations.

B) Left, fewer calculations.C) Either right or left,

same amount of work.

D) None of the above,too many unknowns.

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QUIZ

12. When determining the force in member HG in the

previous question, which one equation of equilibrium is

best to use?

A) MH = 0

B) MG = 0C) MB = 0

D) MC = 0

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