Analysis of Statically Indeterminate Structures using the Force Method, Study notes of Law

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Chapter 4

Analysis of Statically

Indeterminate Structures by

the Force Method

Outline

Statically Indeterminate Structures
Force Method of Analysis: GeneralProcedure
Maxwell’s Theorem of ReciprocalDisplacements; Berti’s Law

4-1 Statically Indeterminate Structures

Advantages & Disadvantages (cont’d)
Although statically indeterminate structure cansupport loading with thinner members & withincreased stability compared to their staticallydeterminate counterpart, the cost savings inmaterial must be compared with the added costto fabricate the structure since often it becomesmore costly to construct the supports & joints ofan indeterminate structure
One has to careful of differential displacementof the supports as well

4-1 Statically Indeterminate Structures

Method of Analysis
To satisfy equilibrium, compatibility & force-displacement requirements for the structure
Force Method
Displacement Method

4-2 Force Method of Analysis:

General Procedure

This will allow the beam to be staticallydeterminate & stable
Here, we will remove the rocker at B
As a result, the load P will cause B to bedisplaced downward as shown in Fig 4.3(b)
By superposition, the unknown reaction at Bcauses the beam at B to be displacedupward, Fig 4.3(c)

4-2 Force Method of Analysis:

General Procedure

Figure 4-

4-2 Force Method of Analysis:

General Procedure

Using methods in Chapter 4 or 3 to solve for ∆ B and f BB

, B

y can be found
Reactions at wall A can then be determinedfrom equation of equilibrium
The choice of redundant is arbitrary

4-2 Force Method of Analysis:

General Procedure

The moment at A, Fig 4.4(a) can bedetermined directly by removing thecapacity of the beam to support moment atA, replacing fixed support by pin support
As shown in Fig 4.4(b), the rotation at Acaused by P is θ A
The rotation at A caused by the redundantM A at A is θ

AA , Fig 4.4(c)

4-2 Force Method of Analysis:

General Procedure

AA A A AA A AA M M α θ α θ

=

0 : requires ity Compatibil ' Similarly,

4-3 Maxwell’s Theorem of Reciprocal

Displacement: Betti’s Law

The displacement of a point B on a structuredue to a unit load acting at point A is equalto the displacement of point A when the loadis acting at point B
Proof of this theorem is easily demonstratedusing the principle of virtual work AB BA

f

f

Example 4.

Determine the reaction at the roller supportB of the beam shown in Fig 4.8(a)
EI is constant
Figure 4-

Example 4.1 - solution

Principle of superposition
By inspection, the beam is staticallyindeterminate to the first degree
The redundant will be taken as B y
Fig 4.8(b) shows application of the principlesuperposition
We assume B y acts upward on the beam

Example 4.

Draw the shear & moment diagrams for thebeam shown in Fig 4.11(a)
EI is constant
Neglect the effects of axial load

Example 4.

Figure 4-