2/6/2011

1

Frameworks - Analysis of Structures

Truss

Members of a truss are slender and notcapable of supporting large lateral loads.Loads must be applied at the joints.

Component Parts

Vertical Bottom Chord

DiagonalEnd Post

Hip Vertical

Deck

Top Chord

Vertical Bottom Chord

DiagonalEnd Post

Hip Vertical

Deck

Top Chord

Support (Abutment)

Trusses Trusses --DefinitionDefinition

Trusses are structures composed of slender members joined together at their end points.

They consists generally of triangular sub-element and are constructed and supported so as to prevent any motion.

What is a Truss?

A structure composed of members connected together to form a rigid framework.

Usually composed of interconnected triangles.

Members carry load in tension or compression.

Standard Truss Configurations

Pratt Parker

Double Intersection Pratt

Howe Camelback

K-Truss

Fink

Warren

Bowstring Baltimore

Warren (with Verticals)

Waddell A Truss Pennsylvania

Double Intersection Warren

Lattice

Pratt Parker

Double Intersection Pratt

Howe Camelback

K-Truss

Fink

Warren

Bowstring Baltimore

Warren (with Verticals)

Waddell A Truss Pennsylvania

Double Intersection Warren

Lattice

2/6/2011

2

Planar TrussPlanar Truss

Planar Trusses - lie in a single plane and all applied loads must lie in the same plane.

Example of Planar Truss: Roof Truss

The roof load is transmitted to the truss at the joints by means of purlins.

Types of Structural Members

Solid RodSolid Bar

Hollow Tube

-Shape

Solid RodSolid Bar

Hollow Tube

-Shape

These shapes are calledcross-sections.

Assumptions for designAssumptions for design

There are four main assumptions made in the analysis of truss

Truss members are connected together at their ends only.

Truss are connected together by frictionless pins.

The truss structure is loaded only at the joints.

The weights of the members may be neglected.

1

2

3

4

Assumptions allow to idealize each truss member as a two-force member (members loaded only at their extremities by equal opposite and collinear forces)

member in compression

member in tension

Connecting pin Gusset plate

A typical truss structure

Joint ConnectionsJoint Connections

The joint connections are usually formed bybolting or welding the ends of the members to acommon plate, called a gusset plate or by simplepassing a large bolt or pin through each of themembers.

2/6/2011

3

Simple TrussSimple Truss

The basic building block of a truss is a triangle. Large truss are constructed by attaching several triangles together A new triangle can be added truss by adding two members and a joint. A truss constructed in this fashion is known as a simple truss.

Truss A truss consists of straight members connected at

joints. No member is continuous through a joint.

Bolted or welded connections are assumed to be pinned together. Forces acting at the member ends reduce to a single force and no couple. Only two-force members are considered.

Most structures are made of several trusses joined together to form a space framework. Each truss carries those loads which act in its plane and may be treated as a two-dimensional structure.

When forces tend to pull the member apart, it is in tension. When the forces tend to compress the member, it is in compression.

ANALYSIS and DESIGN ASSUMPTIONS When designing both the member and the joints of a truss, first it is necessary to determine the forces in each truss member. This is called the force analysisof a truss. When doing this, two assumptions are made:

1. All loads are applied at the joints. The weight of the truss members is often neglected as the weight is usually small as compared to the forces supported by the members.

2. The members are joined together by smooth pins. This assumption is satisfied in most practical cases where the joints are formed by bolting or welding.

With these two assumptions, the members act as two-force members. They are loaded in either tension or compression. Often compressive members are made thicker to prevent buckling.

Simple Trusses

A rigid truss will not collapse under the application of a load.

A simple truss is constructed by successively adding two members and one connection to the basic triangular truss.

Frames Which Cease To Be Rigid When Detached From Their Supports

Some frames may collapse if removed from their supports. Such frames can not be treated as rigid bodies.

A free-body diagram of the complete frame indicates four unknown force components which can not be determined from the three equilibrium conditions.

The frame must be considered as two distinct, but related, rigid bodies.

With equal and opposite reactions at the contact point between members, the two free-body diagrams indicate 6 unknown force components.

Equilibrium requirements for the two rigid bodies yield 6 independent equations.

Analysis of TrussAnalysis of Truss

It has been observed that the analysis of trusscan be done by counting the number memberand joints on the truss to determine the truss isdeterminate, unstable or indeterminate.

2/6/2011

4

What are we looking for?

The support reaction .The force in each member.

How many equations are available? How many unknowns?

Each joint- 2 equationsUnknowns- number of members+ support reaction.

Stability Criteria

Truss stable : m=2j-32j - number of equations to be solved, (j = number of joints);m - number of members;3 - number of support reaction.

m2j-3 Statically indeterminate

Example

m (Number of members) =

m=2j-3

13

j (Number of joints) =

Number of supports=

8

3

Method of Joints Method of Joints --TrussTruss

The truss is made up of single bars, which are either incompression, tension or no-load.

The means of solving force inside of the truss useequilibrium equations at a joint. This method is knownas the method of joints.

Method of Joints Method of Joints --TrussTruss

The method of joints uses the summation offorces at a joint to solve the force in themembers. It does not use the momentequilibrium equation to solve the problem. In atwo dimensional set of equations,

In three dimensions,x y0 0 F F

z 0 F

Analysis of Trusses by the Method of Joints Dismember the truss and create a free-body

diagram for each member and pin.

The two forces exerted on each member are equal, have the same line of action, and opposite sense.

Forces exerted by a member on the pins or joints at its ends are directed along the member and equal and opposite.

2/6/2011

5

Method of Joints

Truss in Equilibrium => Each Joint in Equilibrium

Method of Joints

Truss in Equilibrium => Each Joint in Equilibrium

Problem 1

Determine the force in each member of the truss and state if themembers are in tension or compression. P1 = 800 kN and P2 =400 kN.

Problem 1

Determine the force in each member of the truss and state if themembers are in tension or compression. P1 = 2 kN and P2 =1.5 kN.

Problem 1

Determine the force in each member of the truss and state if themembers are in tension or compression.

ZERO-FORCE MEMBERS

If a joint has only two non-colinear members and there is no external load or support reaction at that joint, then those two members are zero-force members. In this example members DE, CD, AF, and AB are zero force members.You can easily prove these results by applying the equations of equilibrium to joints D and A.

Zero-force members can be removed (as shown in the figure) when analyzing the truss.

2/6/2011

6

ZERO FORCE MEMBERS (continued)

If three members form a truss joint for which two of the members are collinear and there is no external load or reaction at that joint, then the third non-collinear member is a zero force member.Again, this can easily be proven. One can also remove the zero-force member, as shown, on the left, for analyzing the truss further.Please note that zero-force members are used to increase stability and rigidity of the truss, and to provide support for various different loading conditions.

Zero-Force Members

Zero-Force Members Zero-Force Members

Joints Under Special Loading Conditions Forces in opposite members intersecting in

two straight lines at a joint are equal. The forces in two opposite members are

equal when a load is aligned with a third member. The third member force is equal to the load (including zero load).

The forces in two members connected at a joint are equal if the members are aligned and zero otherwise.

Recognition of joints under special loading conditions simplifies a truss analysis.

Analysis of Trusses by the Method of Sections When the force in only one member or the

forces in a very few members are desired, the method of sections works well.

To determine the force in member BD, pass a section through the truss as shown and create a free body diagram for the left side.

With only three members cut by the section, the equations for static equilibrium may be applied to determine the unknown member forces, including FBD.

2/6/2011

7

Method of Sections

Truss in Equilibrium => Each PART in Equilibrium

Method of Sections

Truss in Equilibrium => Each PART in Equilibrium

Efficient when forces of only a few members are to be found

Method of Sections - Procedure

Free Body Diagram

Determine external reactions of entire truss

Decide how to section trussHint: Three(3) unknown forces at the most

Method of Sections Procedure (contd)

Free Body Diagram

Draw FBD of one partHint: Choose part with least number of forces

Method of Sections Procedure (contd)

Free Body Diagram

Establish direction of unknown forces(a)Assume all forces cause tension in member

Numerical results: (+) tension (-) compression

(a)Guess DirectionNumerical results: (+) Guess is correct (-) Force in opposite direction

Method of Sections Procedure (contd)

Equations of Equilibrium

Take moments about a point that lies on the intersection of the lines of action of two unknown forces

0

0

0

M

F

F

y

x

2/6/2011

8

Problem 4

Determine the force in members BC, HC, and HG of the bridgetruss, and indicate whether the members are in tension orcompression.

Trusses Made of Several Simple Trusses Compound trusses are statically

determinant, rigid, and completely constrained.

32 nm

Truss contains a redundant memberand is statically indeterminate.

32 nm

Necessary but insufficient condition for a compound truss to be statically determinant, rigid, and completely constrained,

nrm 2

non-rigid rigid32 nm

Additional reaction forces may be necessary for a rigid truss.

42 nm

Sample Problem

Determine the force in members FH, GH, and GI.

SOLUTION:

Take the entire truss as a free body. Apply the conditions for static equilib-rium to solve for the reactions at A and L.

Pass a section through members FH, GH, and GI and take the right-hand section as a free body.

Apply the conditions for static equilibrium to determine the desired member forces.

Sample Problem

SOLUTION:

Take the entire truss as a free body. Apply the conditions for static equilib-rium to solve for the reactions at A and L.

kN 5.12

kN 200 kN 5.7

m 25kN 1m 25kN 1m 20kN 6m 15kN 6m 10kN 6m 50

A

ALFL

LM

y

A

Sample Problem

Pass a section through members FH, GH, and GIand take the right-hand section as a free body.

kN 13.13

0m 33.5m 5kN 1m 10kN 7.500

GI

GI

H

FF

M

Apply the conditions for static equilibrium to determine the desired member forces.

TFGI kN 13.13

Sample Problem

kN 82.130m 8cos

m 5kN 1m 10kN 1m 15kN 7.50

07.285333.0m 15m 8tan

FH

FH

G

FF

MGLFG

CFFH kN 82.13

kN 371.1

0m 10cosm 5kN 1m 10kN 10

15.439375.0m 8

m 5tan32

GH

GH

L

FF

M

HIGI

CFGH kN 371.1

2/6/2011

9

1. 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 revised truss 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.

H

P

A

B C

W

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

A) 0 B) 1 C) 2

D) 3 E) 4

F F

F

1. Using this FBD, you find that FBC = 500 N.Member BC must be in __________.

A) tension B) compression

C) Cannot be determined

2. For the same magnitude of force to be carried, truss members in compression are generally made _______ as compared to members in tension.

A) thicker

B) thinner

C) the same size

FBD

FBC

B

BY

Space Trusses An elementary space truss consists of 6 members

connected at 4 joints to form a tetrahedron.

A simple space truss is formed and can be extended when 3 new members and 1 joint are added at the same time.

Equilibrium for the entire truss provides 6 additional equations which are not independent of the joint equations.

In a simple space truss, m = 3n - 6 where m is the number of members and n is the number of joints.

Conditions of equilibrium for the joints provide 3nequations. For a simple truss, 3n = m + 6 and the equations can be solved for m member forces and 6 support reactions.