TRUSSES

Trusses

Trusses

Trusses

Trusses A framework composed of members joined at their ends to form a

structure is called a truss. Truss is used for supporting moving or

stationary load. Bridges, roof supports, electrical transmission towers and

other such structures are common example of trusses. When the members

of the truss lie essentially in a single plane, the truss is called a plane

truss .

Fig. 3.1 Triangular truss

Trusses

• In a wide range of applications, trusses and cables are

employed to support transverse loads.

• Members forming the truss and cables are predominantly

subjected to axial loads through external loading is in the

transverse direction.

Trusses Fig. 3.1 shows most basic triangular truss. Members are connected by pin-

joints, which arrest translation but not rotation. Each member has three degrees of freedom in a plane, two translations and one rotation. Total degrees of freedom are 9. Each pin joint arrests two degree of freedom. Hence, degrees of freedom of pin-joint connected structure is 3. For keeping the structure stationary, these three degrees of freedom should be arrested. In the figure, left fixed support arrests two degrees of freedom, whereas in the right, the roller support arrests one degree of freedom. Thus the structure cannot move and the structure is called stable. This type of structure is also called rigid structure .

Trusses Consider a 4-member structure as shown in Fig. 3.2. This type of structure is called non-rigid. Under

the application of loads, the structure can adopt various configurations. Degree of freedom of this structure is one. This because, each member has 3 degrees of freedom. So total degree of freedom is 12. Four pin joints arrest 2 degrees of freedom. In addition, left and right supports arrest 2 and 1 degrees of freedom, respectively. Hence, total degrees of freedom: 12-8-3=1. This is called mechanism

 

Fig. 3.2 4-member structure

Trusses• Two members connected at a joint form a hinged arch , as shown below. A

hinged arch may be added to any stable truss to form another stable truss, as long as the angle of the arch is other than 180º. A truss which can be assembled in this manner is called a simple truss .  

The simple plane truss is built up from an elementary triangle by adding two new members for each new pin. For simple plane truss, the following relation exists: m=2j-3 where m is the number of member and j is the number of pin joints. This can be proven as follows: It is seen that to increase one joint in a rigid truss, two more members are required. The following table gives number of members m versus number of joints

 

TrussesAnother way to prove this relation is as follows. In 2-dimensional space, the equilibrium of each joint is specified by three scalar force equations. There are in total 2j such equations for a plane truss with j joints. For the entire truss composed of m members, there are m unknowns (tensile or compressive forces in the member) plus 3 unknown support reactions in the general case of statically determinate plane truss. Thus, for any plane truss, the equation m+3=2j will be satisfied, if the truss is statically determinate internally.

NOTE: A member of the truss is a two force member. Hence, the forces are collinear. A member of the truss is either in tension or compression.

• It is customary to idealize the joints in trusses as Frictionless pin joints (joints cannot support a moment).

• At joints, members are arrested from translation but rotation motion is allowed.

Idealization of joints

Method of Joints

FGFFGA

FGBFGC

15 kN

Method of Joints

FGFFGA

FGBFGC

15 kN

2FGC sin Ѳ– 15 kN = 0 FGC = (15 X 5)/(2 X 3) = 12.5 kN

Method of Joints

FCG

FCB

FCF

-FCG sinѲ – FCF sinѲ = 0FCF = - FCG = -12.5 kN

- FCB – FCG cosѲ + FCF cos Ѳ = 0FCB = –2FCGcos Ѳ = 10 kN

Method of Joints

FFG

FFCRB,V

∑FV = RB,V +(3/5)FFC = 0 or, RB,V = - (3/5)FFC = + 7.5 kN

∑FH = -(4/5)FFC - FFG = 0 or, FFG = - (4 /5)FFC = 10 kN

Clamped SupportWhile in the pinned support, the member is restrained from translating,

In the clamed support, the member cannot even rotate.

Clamped Support...

• Cannot translate,• Cannot rotate.

No DoF at all. Also called built-in support

P

FxMz

Fy

zx

y Statically determinate

Clamped Support...

L

Method of Joints: At each joint the forces in the members meeting at the joint and the loads

at the joint, if any, constitute a system of concurrent forces. Hence, two independent equations of equilibrium can be formed about each joint. For starting analysis, a joint is selected where there are only two unknown forces. Many a time such a joint can be identified only after finding the reaction at the support by considering the entire frame. Then making use of the two equations of equilibrium for the system of forces acting at the joint those two unknown forces are found. Then the next joint is selected for analysis where there are now only two unknown forces. Thus, the analysis proceeds from joint to joint to find the forces in all the members.  

Let us start with joint C. Equilibrium at joint C Applying Lami's theorem

Joint B has three unknown force. So we select joint D, which has only two unknown forces .

Method of Joints:

Method of Joints:

Method of Joints:

Now joint E has 3 unknown and joint B has two unknown now we select joint B

Resolving the forces in vertical dedication

Method of Joints:

Method of Joints:

Hence, the actual direction is opposite to that show. Thus,

Coming to point E now,

Method of Joints:

Resoling the forces in vertical direction

Thus, the problem has been solved