Fundamentals of Structural Analysis Episode 1 Part 5 pdf

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

75

Work done by external force (left) and work done by interanl force (right)-case 1.

The conservation of mechanical energy principle calls for

2

1 ∑=

n

i

i

1

P ∆i =

2

1

j

M

j

∑Fj

V =1

(15)

(2) The case of virtual unit load acts alone. The figure below illustrates the force￾displacement histories.

Work done by external force (left) and work done by interanl force (right)-case 2.

Again, the energy conservation principle calls for

2

1 (1) ∆o

'

=

2

1

j

M

j

j ∑ f v =1

(16)

(3) The case of the virtual unit load being applied first, followed by the application of the

real loads. The force-displacement histories are shown in the figures below.

∆i

Pi

Disp.

External Force

Vj

Elong.

Internal Force

Fj

∆o

'

1

Disp.

External Force

vj

Elong.

Internal Force

fj

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

76

Work done by external forces (left) and work done by interanl forces (right)-case 3.

Application of the energy conservation principle leads to

2

1 (1) ∆o

'

+

2

1 ∑=

n

i

i

1

P ∆i + (1) (∆o) = 2

1

j

M

j

j ∑ f v =1

+

2

1

j

M

j

∑Fj

V =1

+ j

M

j

∑ f j

V =1

(17)

Substracting Eq. 17 by Eq. 15 and Eq. 16 yields

(1) (∆o) = j

M

j

∑ f j

V =1

(14)

which is the principle of virtual force statement expresssed in the unit-load context.

Example 15. Find the vertical displacement at node 2 of the truss shown, given E=10

GPa, A=100 cm2

for all bars.

1

Disp.

External Force

Elong.

Internal Force ∆o

'

∆o

vj

fj

Vj

∆i

Pi

Disp.

External Force

Fj