Engineering Statics

ENGR 2301

Chapter 6

Analysis of Structures

Application

6 - 2

Design of support structures requires

knowing the loads, or forces, that each

member of the structure will

experience.

TRUSSES

Application Of Truss

One of the most famous structures in the world, the Eiffel

Tower, is a truss – a series of inter-connected triangles.

Trusses are also used to construct bridges, roofs, floors,

and even the Statue of Liberty.

The Eiffel tower was designed by

Gustave Eiffel and the Statue of

Liberty was designed by Frédéric

Bartholdi, with the help of Gustave

Eiffel who designed the interior

structure.

Truss

A technical definition of a truss is, “A rigid framework

composed of members connected at joints and arranged

into a network of triangles.”

Steven Ressler, Ph.D., U.S. Military Academy at West Point

A truss in architecture and engineering is a timber or metal

structural member that is formed of one triangle or a series of

triangles in a single plane. A truss requires less material than a

solid beam in attaining long spans for carrying heavy loads,

making it especially useful in constructing bridges and roofs.

History Of Truss

Some historians have stated that trusses date back to

18th century Greece, and others argue that there are

13th century structures in Italy that contain trusses.

Introduction

6 - 7

• For the equilibrium of structures made of several

connected parts, the internal forces as well the external

forces are considered.

• In the interaction between connected parts, Newton’s 3rd

Law states that the forces of action and reaction

between bodies in contact have the same magnitude,

same line of action, and opposite sense.

• Three categories of engineering structures are considered:

a) Trusses: formed from two-force members, i.e.,

straight members with end point connections and

forces that act only at these end points.

b) Frames: contain at least one one multi-force

member, i.e., member acted upon by 3 or more

forces.

c) Machines: structures containing moving parts

designed to transmit and modify forces.

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 analysis

of 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.

Definition of a Truss

6 - 9

• 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.

Definition of a Truss

6 - 10

Members of a truss are slender and not capable of

supporting large lateral loads. Loads must be applied at

the joints.

Definition of a Truss

6 - 11

Simple Trusses

6 - 12

• 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.

Analysis of Trusses by the Method of Joints

6 - 13

• Dismember the truss and create a freebody

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.

• Conditions of equilibrium are used to solve for

2 unknown forces at each pin (or joint), giving a

total of 2n solutions, where n=number of joints.

Forces are found by solving for unknown forces

while moving from joint to joint sequentially.

• Conditions for equilibrium for the entire truss

can be used to solve for 3 support reactions.

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.

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.

Joints Under Special Loading Conditions

6 - 16

• 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.

Space Trusses

6 - 17

• 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 3n

equations. For a simple truss, 3n = m + 6 and the

equations can be solved for m member forces and

6 support reactions.

Sample Problem 6.1

6 - 18

Using the method of joints, determine

the force in each member of the truss.

SOLUTION:

• What’s the first step to solving this

problem? Think, then discuss this with a

neighbor.

• DRAW THE FREE BODY DIAGRAM

FOR THE ENTIRE TRUSS (always first)

and solve for the 3 support reactions

• Draw this FBD and compare your sketch

with a neighbor. Discuss with each other

any differences.

Sample Problem 6.1

6 - 19

SOLUTION:

• Based on a free body diagram of the entire truss,

solve the 3 equilibrium equations for the reactions

at E and C.

ft 6ft 12lb 1000ft 24lb 2000

0

E

MC

lb 000,10E

xx CF 0 0xC

yy CF lb 10,000 lb 1000 - lb 20000

lb 7000yC

• Looking at the FBD, which “sum of moments”

equation could you apply in order to find one of

the unknown reactions with just this one equation?

• Next, apply the remaining

equilibrium conditions to

find the remaining 2 support

reactions.

Sample Problem 6.1

6 - 20

534

lb 2000 ADAB FF

CF

TF

AD

AB

lb 2500

lb 1500

FDB FDA

FDE 235 FDA CF

TF

DE

DB

lb 3000

lb 2500

• We now solve the problem by moving

sequentially from joint to joint and solving

the associated FBD for the unknown forces.

• Which joint should you start with, and why?

Think, then discuss with a neighbor.

• Joints A or C are equally good because each

has only 2 unknown forces. Use joint A and

draw its FBD and find the unknown forces.

• Which joint should you move to next, and why? Discuss.

• Joint D, since it has 2 unknowns remaining

(joint B has 3). Draw the FBD and solve.

Sample Problem 6.1

6 - 21

• There are now only two unknown member

forces at joint B. Assume both are in tension.

lb 3750

25001000054

54

BE

BEy

F

FF

CFBE lb 3750

lb 5250

375025001500053

53

BC

BCx

F

FF

TFBC lb 5250

• There is one remaining unknown member

force at joint E (or C). Use joint E and

assume the member is in tension.

lb 8750

37503000053

53

EC

ECx

F

FF

CFEC lb 8750

Sample Problem 6.1

6 - 22

• All member forces and support reactions are

known at joint C. However, the joint equilibrium

requirements may be applied to check the results.

checks 087507000

checks 087505250

54

53

y

x

F

F