deflection of truss [compatibility mode]

8/18/2019 Deflection of Truss [Compatibility Mode]

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Structural Analysis IDEFLECTIONS OF TRUSS

Daniel Rumbi Teruna

School of Civil EngineeringUniversity of North Sumatera


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Principle of Virtual Force-Unit Load Method

In equation this principle is expressed as

δW =δU

δW = work done by external virtual force(s) upon a real displacement system.

δU = work done by internal virtual forces upon a real displacement system.

where

Consider a truss system, represented by the rectangular boxes in the figures below,

loaded by two different loading systems – a real load system and a virtual unit load.

The real load system is the actual load applied to the truss under consideration. Theunit load is a virtual load of unit magnitude applied at a point whose displacement

we want and applied in the direction of the displacement we want. These loads are

shown outside of the boxes together with the displacements under the loads. The

corresponding internal member force and member elongation are shown inside of

the boxes.


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A real system (left) and a virtual system (right).


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We will now consider three loading cases:

(1) The case of externally applied loads act alone. The applied loads, Pi, and

internal member forces, F j , generate work according to the force-displacement

histories shown in the figures below.


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The conservation of mechanical energy principle calls for

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

(1)


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

forcedisplacement histories.

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

Again, the energy conservation principle calls for

(2)


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(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.


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Application of the energy conservation principle leads to

Substracting Eq. 3 by Eq. 1 and Eq. 2 yields

(3)

(4)

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

Example 1. Find the vertical displacement at node 2 of the truss shown, given E =10GPa, A=100 cm2 for all bars.


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Solution. Using the unit load method requires

the solution for the member elongation, Vi,

under the appied load and the virtual member

force, fi, under the unit load as shown in the

figure below

A unit load applied in the direction of the displacement to be solved.


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Thus, the vertical displacement at node 2 is 0.0533 mm, downward.

Example 2. Find (a)the relative movement of nodes 2 and 6 in the direction joining

them and (b) rotation of bar 2, given E =10 GPa, A=100 cm2 for all bars.


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Solution. The nodal displacments

related to the relative movement androtation in question are depicted in

the figures below.

Relavant nodal displacements.

To find the relative movement between node 2 and node 6, we can apply a pair of unit

loads as shown. We shall call this case as case (a).


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Unit load for movement between node 2 and

node 6 in the direction of 2-6, case (a).

Unit load to find rotation of bar 2, case (b).

To find the rotation of bar 2, we can apply a pair of unit loads as shown. We shall

call this case as case (b).


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For case (a), Eq. 4 becomes

The relative movement in the direction of 2-6 is 8.86 mm in the opposite direction of

what was assumed for the unit load, i.e., away from each other, not toward each other.

For case (b), Eq. 4 becomes

For the rotation of bar 2, we note that the –9.83 computed represents a relative vertical

movement between node 2 and node 3 of 9.83 mm in the opposite direction of what

was assumed for the pair of unit loads. That relative vertial movement translates into a

counterclockwise rotation of 9.83mm/4,000mm=0.0025 radian.


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deflection of truss [compatibility mode]

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