in the previous chapter we have employed the...
TRANSCRIPT
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.