Structural Analysis - EMUcivil.emu.edu.tr/courses/civl211/LECTURE-9.pdf · roof-supporting truss ... • For design and analysis of a truss, we need to obtain the ... Example T: Example:

Structural Analysis

Chapter Objectives

• To show how to determine the forces in the

members of a truss using:

the method of joints and

the method of sections.

Chapter Outline

• Two-force members

• Planar (Simple) Trusses

• The Method of Joints

• Zero-Force Members

• The Method of Sections

Two-Force Members

• When a member is subjected to

no couple moments

and

forces are applied at:

only two points on a member,

then the member is called

a two-force member

Two-Force Members Example

• Forces at A and B are summed to obtain their respective resultants FA and FB

• These two forces will maintain translational and force equilibrium provided FA is of equal magnitude and opposite direction to FB

• Line of action of both forces is known and passes through A and B

When any member is subjected to:

No couple moments

and

TWO EQUAL,

OPPOSITE and

COLLINEAR



then this member is called

a two-force member and this member is in equilibrium.

A B

F F

A B

F F

two-force member (in equilibrium)

two-force member (in equilibrium)

A B

F

F

NOT a two-force member (NOT in EQUILIBRIUM)

A B

F F

M0

A B

F

F

A B

F

F

A B M0

M0

A B

F F

When any member is subjected to:

No couple moments

and

TWO EQUAL,

OPPOSITE and

COLLINEAR



then this member is called

a two-force member and this member is in equilibrium.






Two-Force Members

PLANAR Trusses

OR

2-D Trusses

SIMPLE Trusses

OR

• A simple truss is constructed starting with a basic triangular element such as ABC and connecting two members (AD and BD) to form an additional element.

Simple Trusses

• To prevent collapse, the form of a truss must be rigid

• The four bar shape ABCD will collapse if a diagonal member AC is not added

• The simplest form that is rigid or stable is a triangle

Simple Trusses

ALL the truss members,

• member-end forces and

• applied forces are in one plane (2D),

(in the same plane).

That is why it is called PLANAR or 2-D

TRUSSES.

Simple Trusses

•A truss is a structure composed of slender

members joined together at their end points

JOINT (NODE)

MEMBER

Simple Trusses

• Joint connections are formed

by bolting or welding the ends

of the members to a common

plate, called a gusset plate, or

by simply passing a large bolt

or pin through each of the

members

Simple Trusses

• Planar trusses lie on a single plane and are used to support roofs and bridges.

• The truss ABCDE shows a typical roof-supporting truss

• Roof load is transmitted to

the truss at joints by

means of a series of

purlins, such as DD’

Simple Trusses

Simple Trusses

• The loads transmitted at the joints are resisted by the members of the truss and forces are developed at the ends of each member.

• Finally all the loads transmitted to the truss are in turn transmitted to the supports and then to the supporting soil.

Simple Trusses

Simple Trusses

K Bridge Truss

Warren Roof Truss

• The loads transmitted at the joints are resisted by the members of the truss and forces are developed at the ends of each member

Member-end force

Member-end force

•forces are developed at the

ends of each member is called

the Member-end force

•Each member-end force is either

TENSION

or

COMPRESSION

Simple Trusses

• If the force tends to elongate the member, it is a

tensile force,

• If the force tends to shorten the member, it is a

compressive force.

Simple Trusses

Each member is in equilibrium under the action of

only two forces that are:

- equal in magnitude,

- opposite in direction and

- collinear.

Therefore truss members are called:

TWO-FORCE MEMBERS

Simple Trusses

• Analysis of Planar Trusses means:

Find ALL the member-end forces

developed in each member,

Find the reactions at the supports

Simple Trusses

Assumptions for Analysis and Design

1. “All loadings are applied at the joints”

• Assumption is true for most of the applications of bridge and roof trusses

• Weight of the members are neglected since the forces supported by the members are large in comparison

• If member’s weight is considered, apply it as a vertical force, by giving half of the magnitude to each end of the member.

Simple Trusses

Assumptions for Analysis and Design

2. “The members are joined together by smooth pins”

• Assumption true when bolted or welded joints are used, provided the center lines of the joining members are concurrent

smooth pin

Simple Trusses

• Each truss member acts as a two force member,

therefore the forces at the ends must be directed

along the axis of the member.

• If the force tends to elongate the member, it is a

tensile force,

• If the force tends to shorten the member, it is a

compressive force.

Simple Trusses

• Important to state the nature of the force in the actual design of a truss: tensile or compressive.

• Compression members must be made thicker than tensile member to account for the buckling or column effect during compression.

Simple Trusses

Action of member-end forces on the pins at

the joints: EQUAL and OPPOSITE force

is applied on the pin

Pin at joint C

The Method of Joints • For design and analysis of a

truss, we need to obtain the force in each of the members.

• Dis-assemble the truss and draw the FBD of each pin and each member.

• Considering the equilibrium of a pin at a joint of the truss, a member-end force becomes an external force on the pin at a joint and equations of equilibrium can be applied.

• This forms the basis for the method of joints

2

3

1

AY

AX

CY Action of member-end forces on the pins

at the joints: EQUAL and OPPOSITE

force is applied on the pin

Pin at joint C

y

x

The Method of Joints

• The force system acting at each pin at a joint is coplanar and concurrent

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

∑Fx = 0 and ∑Fy = 0

∑Fx = 0 and ∑Fy = 0 ∑Fx = 0 and ∑Fy = 0


Method of Analyses 1) If x and y coordinates are not given show it on FBD.

This will help to obtain the x and y components of all the forces acting on that joint.

2) Assume all the unknown forces acting on the members to be in TENSION.

3) Draw FBD for each joint and attempt to solve it by using Equilibrium equations. To start with select that joint that has:

- at least (minimum) one known force and

- at most (maximum) two unknown forces.

4) If the joint has a support reaction you can find it earlier and then apply method of Joint

SOLUTION BY METHOD OF JOINT

Method of Analyses

5) Apply equilibrium equations ∑Fx = 0 and ∑Fy = 0

6) Solve the unknown forces based on that joints FBD.

7) APPLY THE SAME METHODOLOGY FOR THE REST OF THE JOINTS.

SOLUTION METHOD OF JOINT

Example:

1 2

3

Determine the force in each member of the truss and

state if the members are in tension or in compression.

Use method of joints. y

x


Ay

1. Determine the forces in all the members of the truss and state

whether they are in tension T or compression C .

•use the method of joints.

•give your results in tabular form

2. Find the reactions at the supports A and E.

Example:

y

x

P2

P1 = 4.355 kN

P2 = 5.276 kN

= 18.6754

D

P1

2 m 3.5 m 3.5 m

5.1

25

m

1

2 3 4

5

6 7

A

B C

E


The reactions and member forces of the truss:

type force member

T 2.17 kN F1

T 5.017 kN F2

C -5.017 kN F3

C -4.355 kN F4

C -1.813 kN F5

T 5.659 kN F6

T 5.659 kN F7

Ax= -5 kN Ay= -4.143 kN Ey= 10.187 kN

Example:

1. Find the reactions at the pin supports A and B. 2. Determine the forces in all the members of the truss and state

whether they are in tension (T) or compression (C). Use the method of joints. Give your results in tabular form.

Note that members 2 and 6 are collinear!

1 2

4

3

6

5

y

x


The reactions and member forces of the truss:

type force member

T 1200 kN F1

C -1342 kN F2

---- 0 kN F3

T 2100 kN F4

C -1273 kN F5

C -1342 kN F6

Ax= 0 Ay= -2100 kN Bx= -1500 kN By=2100 kN

1. Find the reactions at the pin supports A and B. 2. Determine the forces in all the members of the truss and state


Note that member 1 and 5 are NOT COLLINEAR!

y

x


2.8

5 m

Example T:

Example: The Method of Joints

1. Find the reactions at the pin supports A and G. 2. Determine the forces in all the members of the truss and state


12 kN 7 kN

2.5 m 3.2 m 3.2 m 2.5 m

3.5 m

2.1 m

A B C E G

L M N 6 kN

y

x

Example: The Method of Joints

Finding the effective forces acting on the joints L and M

12 kN 7 kN

1.1 m

3.2 m

7.875 kN 11.125 kN

3.2 m

L L M M

2.1 m

∑Fy = -7-11= -19 kN ∑ML = - 35.6 kN m

Fat M = - 35.6/3.2 = - 11.125 kN So Fat L = -19 – (- 11.125) = -7.875 kN

Example: The Method of Joints y

x

11.125 kN

2.5 m 3.2 m 3.2 m 2.5 m

3.5 m

A B C E G

L M N 6 kN

7.875 kN

27.5°

51.6°

11.2 kN

46 kN

8.5 kN

1.85 m

1.60 m 0.75 m

1.6

0 m

2

.10

m

C D

A

B E F

1 2

3

4 5

6

7

8

9

1. Find the reactions at the pin supports A and F. 2. Determine the forces in all the members of the truss and state


Example T: The Method of Joints

ZERO-FORCE Members

• Method of joints is simplified when the members which support no loading are determined.

Within a truss, zero-force member exists due to:

• Shape of the member connections

• Support reaction types.

ZERO-FORCE Members

Shape of the truss

• Zero-force members (support no loading )

are used to increase the stability of the

truss during construction and to provide support if the applied loading is changed.

Shape of the truss

• Consider the truss shown

• From the FBD of the pin at point A,

members AB and AF become

zero force members

ZERO-FORCE Members

ZERO-FORCE Members

Shape of the truss

• Zero-force members (support no loading )

are used to increase the stability of the

truss during construction and to provide support if the applied loading is changed.

0 0

Shape of the truss

• Consider FBD of joint D

• DC and DE are zero-force members.

ZERO-FORCE Members

Shape of the truss

• Consider FBD of joint D

• DC and DE are zero-force members.

ZERO-FORCE Members

0 0

• General rule 1:

IF only two members form a truss joint and no external load or support reaction is

applied to the joint,

THEN BOTH members must be zero-force members.

ZERO-FORCE Members

Shape of the truss

0 0

0

0

Consider the truss shown:

• From the FBD of the pin of the joint D,

DA is a zero-force member

• From the FBD of the pin of the joint C,

CA is a zero-force member

ZERO-FORCE Members

Shape of the truss

Consider the truss shown:

• From the FBD of the pin of the joint D,

DA is a zero-force member

• From the FBD of the pin of the joint C,

CA is a zero-force member

ZERO-FORCE Members

Shape of the truss

0 0

• General Rule 2:

IF three members form a truss

joint for which two of the members are

collinear, also having the same

magnitude and sign;

THEN the third member is a zero-force

member provided no external force or

support reaction is applied to the joint

ZERO-FORCE Members

Shape of the truss

ZERO-FORCE Members

Shape of the truss

FOR ANY JOINT

IF

1- NO SUPPORT

2- NO EXTERNAL LOADING

ONLY TWO MEMBERS

BOTH ARE ZERO MEMBERS

ONLY THREE MEMBERS

BUT

TWO OF THEM ARE COLLINEAR

THE REMAINING MEMBER

IS ZERO MEMBER

ZERO-FORCE Members

Shape of the truss

0

ZERO-FORCE Members

Shape of the truss

Example:

Determine all the zero-force members of the Fink roof truss. Assume all joints are pin connected.

ZERO-FORCE Members

Shape of the truss

0

ZERO-FORCE Members

Shape of the truss

0 0

ZERO-FORCE Members

Shape of the truss

0 0 0

ZERO-FORCE Members

Shape of the truss

ZERO-FORCE Members

Shape of the truss

0

ZERO-FORCE Members

Shape of the truss

0 0

ZERO-FORCE Members

Shape of the truss

0 0 0

ZERO-FORCE Members

Shape of the truss

0 0 0 0

ZERO-FORCE Members

Shape of the truss

0 0 0 0

0

ZERO-FORCE Members

Shape of the truss

0 0 0 0

0

0

ZERO-FORCE Members

Shape of the truss

0

0 0 0 0 0 0

ZERO-FORCE Members

Shape of the truss

ZERO-FORCE Members

Shape of the truss

0

ZERO-FORCE Members

Shape of the truss

0

0

0

ZERO-FORCE Members

Shape of the truss

0

-FCD = Dy

ZERO-FORCE Members

Support types

4 m

10 kN B

A

C

D

5 kN 2 kN

4 m

10 kN B

A

C

D

5 kN

Dy

2 kN

4 m

10 kN B

A

C

D

5 kN

Dy

2 kN

-FAD = Dx

ZERO-FORCE Members

Support types

4 m

10 kN B

A

C

D

5 kN

2 kN

4 m

10 kN B

A

C

D

5 kN

Dx

2 kN

0

4 m

10 kN B

A

C

D

5 kN

Dx

2 kN

No zero member

-FCD = Dy

-FAD = Dx

ZERO-FORCE Members

Support types

4 m

10 kN B

A

C

D

5 kN

2 kN

4 m

10 kN B

A

C

D

5 kN

Dy

Dx

2 kN

No zero member

ZERO-FORCE Members

Support types

4 m

10 kN B

A

C

D

5 kN

2 kN

4 m

10 kN B

A

C

D

5 kN

Ay

2 kN

4 m

10 kN B

A

C

D

5 kN

4 m

10 kN

B

A

C

D

5 kN

Ay

0 -FAB = Ay

Dy

Dx = 0

-FCD = Dy

ZERO-FORCE Members

Support types

0

4 m

4 m

4 m

5 kN

10 kN

10 kN

4 m

B

A

C

D

E

F

G

5 kN

ZERO-FORCE Members

Shape of the truss

4 m

4 m

4 m

5 kN

10 kN

10 kN

4 m

B

A

C

D

E

F

G

5 kN

0 0

ZERO-FORCE Members

Shape of the truss

4 m

4 m

4 m

5 kN

10 kN

10 kN

4 m

B

A

C

D

E

F

G

5 kN

0 0

ZERO-FORCE Members

Shape of the truss

0

0 0

0

ZERO-FORCE Members

Shape of the truss

4 m

4 m

4 m

5 kN

10 kN

10 kN

4 m

B

A

C

D

E

F

G

5 kN

0

Summary

Truss Analysis

• A simple truss consists of triangular

elements connected by pin joints.

• The force within determined by assuming

all the members to be two force member,

connected concurrently at each joint.

Summary

Method of Joints

• For a coplanar truss, the concurrent force at

each joint must satisfy force equilibrium

• For numerical solution of the forces in the

members, select a joint that has FBD with at

most 2 unknown and 1 known forces

Summary

Method of Joints

• Once a member force is determined, use its

value and apply it to an adjacent joint.

• Forces that PULL the joint are in Tension.

• Forces that PUSH the joint are in Compression.

Summary

Method of Joints

• To avoid simultaneous solution of two

equations, sum the force in a direction that

is perpendicular to one of the unknown.

• To simplify problem-solving, first identify all

the zero-force members.

Structural Analysis - EMUcivil.emu.edu.tr/courses/civl211/LECTURE-9.pdf · roof-supporting truss ... • For design and analysis of a truss, we need to obtain the ... Example T: Example:

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