Advanced Methods of Structural Analysis

6.2 Initial Parameters Method 147

Figure6.1c shows the frame due to action of horizontal forceP. At fixed support

Athe linear and angular displacements are zero, while at pinned supportBthe angle

of rotationB¤ 0. The jointsCandDhave the horizontal displacementsCand

D; under special assumptions these displacements are equal. JointsCandDhave

angular displacementsCandD(they are not labeled on the sketch). The linear

and angular displacements of jointsCandDlead to deformation of the vertical

members as shown on the sketch. Since supportAis fixed, then the left memberAC

has an inflection point.

Figure6.1d shows the frame with hinged ends of the cross-barCD; the frame is

subjected to horizontal forceP:In this case the cross-bar and columnBDhas adis-

placementbut does not havedeflectionand members move as absolutely rigid one –

the motion of the memberCDis a translation, while the memberBDrotates around

pointB. Thus, it is a possible displacement of the member without the relative dis-

placements of its separate points. So a displacement is not always accompanied by

deflections, however, deflections are impossible without displacement of its points.

Figure6.1e, f shows the shapes of the beams caused by settlement of support. In

case 6.1e, a new form of statically determinate beam is characterized by displace-

ment of portionH-Bas absolutely rigid body, i.e., withoutdeflectionof the beam.

In case 6.1f, a new form of the beam occurs with deflection of the beam.

There are two principle analytical approaches to computation of displacements.

The first of them is based on the integration of the differential equation of the elas-

tic curve of a beam. Modification of this method leads to the initial parameters

method. The second approach is based on the fundamental energetic principles. The

following precise analytical methods represent the second group: Castigliano theo-

rem, dummy load method (Maxwell–Mohr integral), graph multiplication method

(Vereshchagin rule), and elastic load method.

All methods from both groups are exact and based on the following assumptions:

1.Structures are physically linear (material of a structures obey Hook’s law);
2.Structures are geometrically linear (displacements ofa structures are much less
than their overall dimensions).

6.2 Initial Parameters Method

Initial parameters method presents a modification of double integration method in

case when a beam has several portions and as result, expressions for bending mo-

ments are different for each portion. Initial parameter method allows us to obtain an

equation of the elastic curve of a beam with any type of supports (rigid or elastic)

and, most important, for any number of portions of a beam.

Fundamental difference between the initial parameter and the double integration

method, as it will be shown below, lies in the following facts:

1.Initial parameters method does not require setting up the expressions for bending
moments for different portions of a beam, formulating corresponding differ-
ential equations and their integration. Instead, the method uses a once-derived