Struct2 Lecture Notes #3

8/11/2019 Struct2 Lecture Notes #3

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Notes in Structural Analysis II Glenn M. Pintor / De La Salle University

Analysis of Statically Indeterminate Structures: Force Method, Bettis & Maxwells Theorems

Method of Analysis for Statically Indeterminate Structures

! Satisfy equilibrium, compatibility, and force-displacement requirements

o Equilibrium reactive forces must hold structure at rest

o Compatibility various segments fit together without intentional breaks

o force-displacement linear elastic response of material

! Force method of analysis: General procedure

1. choose one of the support reactions as redundant, temporarily remove its effect on the

primary structure2. determine the expected displacementprimary structureon the due to effects of external

loads3.

by principle of superposition, apply the displacement due to the redundantexternal

4. establish the compatibility condition/equations relating the displacements in steps 2 & 35.

determine the redundantand/or internal loading/stresses as needed


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! Method of Virtual Work: Trusses

o 1 external virtual unit load acting on the truss joint in the stated direction of !

o

n internal virtual normal force in the truss member caused by the external virtual unitload

o ! external joint displacement caused by real loads on the truss

o N internal normal force in the truss member caused by the real loads

o L length of the member

o A cross-sectional area

o E modulus of elasticity

" External virtual unit loads creates internal virtual forces

" Real loads cause the truss joints to be displaced !in the same direction as the

virtual unit load

"

Each member is displacedNL/AEin the same direction as its respective n force" External virtual work 1!!equals the internal virtual work (or internal virtual

strain) in all truss members

! Maxwells Theorem of Reciprocal Displacements

o The displacement of a point B on a structure due to a unit load acting at point A is equal

to the displacement of point A when the unit load is acting at point B.

!

Indeterminate Truss, Example No. 1

Using the method of consistent deformations, determine the vertical and horizontal reactions at A

and E and the resulting member loads for the truss in the accompanying figure.

The member properties are A = 2 in2and E = 29x103ksi.

Figure 1 - Truss structure to analyze

"

determine the degree of indeterminacy

The structure is indeterminate to the first degree (r=4, e=3, n=r-n=4-3=1).


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" select redundant and remove restraint

To solve for the single degree of indeterminacy, the structure has to be reduced to a statically

determinate and stable structure. This is done by removing a redundant support.

In this example, the horizontal reaction at support E is selected as a redundant to remove in order to

obtain a primary determinate structure.

Figure 2 - Primary determinate structure

" determine reactions and member forces

Calculate the support reactions of the primary structure, and then determine the individual member

forces by using either the method of sections, or the method of joints.

Figure 3 - Support reactions and member forces

"

calculate deformation at redundant

Use the virtual work method, calculate the horizontal translation of support E that corresponds to

the removed redundant XE. Remove all loads and apply a unit force in the direction of the

redundant.

Figure 4 - Primary structure with unit load applied


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" solve consistent deformation equations

The deflection of the released structure under applied loads, E0.

note: for trusses, only the members which are subjected to the unit load need to be included

Member n(k) N(k) L(ft) nNL/AE (k2

-ft)/AE

AB 1 -25.83 4 -103.33/AE

BC 1 -25.83 4 -103.33/AE

CD 1 -44.167 4 -176.67/AE

DE 1 -44.167 4 -176.67/AE

Total = E0= -560/AE

The deflection of the released structure at E caused by the unit load, fee.

Member n (k) N (k) L(ft)nNL/AE (k

2-

ft)/AE

AB 1 1 4 4/AE

BC 1 1 4 4/AE

CD 1 1 4 4/AE

DE 1 1 4 4/AE

Total =fee= 16/AE

Using equation (1), solve for XE:

-560 (ft2-k)/AE + 16 (ft2-k)/AE*XE= 0

XE= 35

Multiply the unit load by this value to get the final reaction. The positive answer indicates that the

reaction is in the direction of the applied unit load.

"

determine support reactions

Impose the value of the calculated XEalong with the other applied loads on the original truss.

Calculate the remaining reactions using the three static equilibrium equations, ("Fx= 0, "Fy= 0

and "M = 0).

Figure 6 - Truss with resultant support reactions


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The resulting member forces are now determined by using the method of sections, or method of

joints.

However, a much easier method is to use superposition and add the effects caused by the redundant

load on the released structure.

In this example, the 35 k load will cause only the top chord of the truss to experience a 35 k tensile

load in each member. This result can be added to those found in figure 3 above. The final results,


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! Virtual work method, Determinate Truss, Example No. 2


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Struct2 Lecture Notes #3

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