In the previous chapter we have employed the equations

of equilibrium in order to determine the support/joint

reactions acting on a single rigid body.

Determination of the support/joint reactions is only the

first step of the analysis in engineering structures. From

now we will focus on the determination of the forces

internal to a structure, that is, forces of action and reaction

between the connected members.

To determine the forces internal to an engineering

structure, we must dismember the structure and analyze

separate free body diagrams of individual members or

combination of members which are mostly connected each

other using smooth pin connections.

Joint reactions always occur in pairs that are equal in

magnitude and opposite in direction. If not isolated from the

rest of the structure or the environment by means of a free

body diagram, joint forces are not included in the diagram

since they will be internal forces.

In order to determine the joint reactions, the structures must

be separated into at least two or more parts. At these

separation points the joint reactions become external forces

and thus, are included in the equations of equilibrium.

In this chapter trusses, frames and machines will be

examined as engineering structures.

A framework composed of members joined at their ends to form a rigid structure is

called a “truss”. Bridges, roof supports, derricks, grid line supports, motorway

passages and other such structures are examples of trusses.

Structural members commonly used are I-beams,

channels, angles, bars and special shapes which are

fastened together at their ends by welding, riveted

connections, or large bolts or pins using large

plates named as “gusset plates”.

I-Beam (I-Kiriş) Channel Beam (U-Profil)

Angled Beam

(Köşebent – L profil)

Bar (Çubuk))A

Gusset Plate (Bayrak)

The basic element of a plane truss is the triangle. Three

bars joined by pins at their ends constitute a rigid

frame. In planar trusses all bars and external forces

acting on the system lie in a single plane.

P A

B C

A Typical Roof Truss

A

B CD

E

F

G

Support Reactions Support Reaction

External Force

Member(Çubuk)

Joint(Düğüm)

A truss can be extended by additing extra tirangles to the system. Such trusses comprising

only of trianges are called “simple trusses – basit kafes”. In a simple truss it is possible to

check the rigidity of the truss and whether the joint forces can be determined or not by using

the following equation:

m : number of members j : number of joints

m=2j - 3 should be satisfied for rigidity

1. In a truss system it is assumed that all bars are two forces members.

The weights of members are neglected compared to the forces they are

supporting. Therefore members work either in tension or compression.

(Çeki) (Bası)

2. When welded or riveted connections are used to join structural

members, we may usually assume that the connection is a pin joint if the

centerline of the members are concurrent at the joint. In this case the joint

does not support any moment since it allows for the rotation of the

members.

3. It is assumed in the analysis of simple trusses that all external forces are

applied at the pin connections.

4. Since bars used in trusses are long, slender elements they can support

very little transverse loads or bending moments.

Determination of Zero-Force Members (Boş Çubukların Belirlenmesi)

Determination of the zero-force members beforehand will generally facilitate the solution of the problem

1.Rule: When two collinear members are under compression, it

is necessary to add a third member to maintain alignment of the

two members and prevent buckling. We see from a force

summation in the y direction that the force F3 in the third

member must be zero and from the x direction that F1=F2. This

conclusion holds regardless of the angle q and holds also if the

collinear members are in tension. If an external force in y

direction were applied to the joint, then F3 would no longer be

zero.

2. Rule : When two noncollinear members are joined as shown,

then in the absence of an externally load at this joint, the forces

in both members must be zero, as we can see from the two

force summations.

F1 and F2 , F3 and F4 collinear

Equal Force Members (Eşit Yük Taşıyan Elemanlar)

When two pairs of collinear members are joined as shown, the forces in each pair

must be equal and opposite.

1

This method for finding the forces in the members of a truss

consists of satisfying the conditions of equilibrium for the forces

acting on the connecting pin of each joint. The method therefore

deals with the equilibrium of concurrent forces, and only two

independent equilibrium equations are involved (SFx=0, SFy=0).

Sign convention (İşaret anlaşması): It is initially assumed

that all the members work in tension. Therefore, when the

FBDs of pins are being constructed, members are shown

directed away from the joint. After employing the equations of

equilibrium, if the result yields a positive value (+), it means

that the member actually works in tension (T) (çeki), if the

result yields a negative value (-), it means that the member

works in compression (C) (bası).

Example: Determine the Zero-Force Members in the plane truss. Also find the forces in

members EF, KL and FL for the Fink truss shown by the Method of Joints.

2) METHOD OF SECTIONS (Kesim Yöntemi)

When analyzing plane trusses by the method of joints, we need only two of the three

equilibrium equations because the procedures involve concurrent forces at each joint. We can

take advantage of the third or moment equation of equilibrium by selecting an entire section

of the truss for the free body in equilibrium under the action of nonconcurrent system of

forces.

The Method of Sections is often employed when forces in limited number of members are

asked for and is based on the two dimensional equilibrium of rigid bodies (SFx=0, SFy=0,

SM=0). This method has the basic advantage that the force in almost any desired member

may be found directly from an analysis of a section which has cut that member. Thus, it is not

necessary to proceed with the calculation from joint to joint until the member in question has

been reached. In choosing a section of the truss, in general, not more than three members

whose forces are unknown should be cut, since there are only three available independent

equilibrium relations.

Once a truss is cut into two parts, one of the parts is taken into consideration and all the

internal forces now become external from where the cut was passed. The forces are

initially assumed as working in tension, so, they are shown directed away from the

FBD. After employing the equations of equilibrium, if the result yields a positive value

(+), it means that the member actually works in tension (T) (çeki), if the result yields a

negative value (-), it means that the member works in compression (C) (bası).

Before starting to solve with method, if necessary the support reactions can be

determined from the FBD of the whole truss and also zero-force members can be

identified. It is very important to recognize that, only the forces acting on the part are

considered, the forces acting on the other part, which is not considered, should not be

included. The moment center can be any point on or out of the part in consideration.

Example: Determine the force in member BE by the Method of Section.