Fundamentals of Structural Analysis Episode 1 Part 4 pptx

Truss Analysis: Force Method, Part I by S. T. Mau

55

Problem 2. Solve for the force in the marked members in each truss shown.

(1-a) (1-b)

(2)

(3)

(4-a) (4-b)

Problem 2.

4m

4@3m=12m

1 kN

a b 4m

4@3m=12m

1 kN

a b

12 kN

4 @ 4m=16 m

2m

3m

a b

c

6@3m=18m

4m

4m

12 kN

a b

c

3@3m=9m

4m

4m

15 kN

a

A B

b c

3@3m=9m

4m

4m

15 kN a

A B

b

Truss Analysis: Force Method, Part I by S. T. Mau

56

4. Matrix Method of Joint

The development of the method of joint and the method of section pre-dates the advent of

electronic computer. Although both methods are easy to apply, it is not practical for

trusses with many members or nodes especially when all member forces are needed. It

is, however, easy to develop a matrix formulation of the method of joint. Instead of

manually establishing all the equilibrium equations from each joint or from the whole

structure and then put the resulting equations in a matrix form, there is an automated way

of assembling the equilibrium equations as shown herein.

Assuming there are N nodes and M member force unknowns and R reaction force

unknowns and 2N=M+R for a given truss, we know there will be 2N equilibrium

equations, two from each joint. We shall number the joints or nodes from one to N. At

each joint, there are two equilibrium equations. We shall define a global x-y coordinate

system that is common to all joints. We note, however, it is not necessary for every node

to have the same coordinate system, but it is convenient to do so. The first equilibrium

equation at a node will be the equilibrium of forces in the x-direction and the second will

be for the y-direction. These equations are numbered from one to 2N in such a way that

the x-direction equilibrium equation from the ith node will be the (2i-1)th equation and

the y-direction equilibrium equation from the same node will be the (2i)th equation. In

each equation, there will be terms coming from the contribution of member forces,

externally applied forces, or reaction forces. We shall discuss each of these forces and

develop an automated way of establishing the terms in each equilibrium equation.

Contribution from member forces. A typical member, k, having a starting node, i, and

an ending node, j, is oriented with an angle θ from the x-axis as shown.

Member orientation and the member force acting on member-end and nodes.

The member force, assumed to be tensile, pointing away from the member at both ends

and in opposite direction when acting on the nodes, contributes to four nodal equilibrium

equations at the two end nodes (we designate the RHS of an equilibrium equation as

positive and put the internal nodal forces to the LHS):

(2i-1)th equation (x-direction): (−Cosθ )Fk to the LHS

(2i)th equation (y-direction): (−Sinθ )Fk to the LHS

(2j-1)th equation (x-direction): (Cosθ )Fk to the LHS

k

i

j

x

y

θ

k

i

j

θ

Fk

Fk

θ

Fk

Fk

θ i

j