Advanced Methods of Structural Analysis

3.5 Special Types of Trusses 67

Construction of the influence line forHis presented in the table below.

PD 1 left at SPLC PD 1 right at SPLC

H!

P

MCrightD 0 !RB4dCHfD0; H!

P

MCleftD 0 !RA4dCHfD0;

HD

4d

f

RB!IL.H /D

4d

f

IL.RB/ HD

4d

f

RA!IL.H /D

4d

f

IL.RA/

The left portion of the influence line forH(portionA-7) presents the influence

line forRBmultiplied by coefficient4d=fand the right-hand portion (portion

C-B) presents the influence line forRAmultiplied by the same coefficient. The

connecting line is between points 7 andC(Fig.3.23). The negative sign for thrust

indicates that all members of the arched chain are in compression.

ForceVn

Equilibrium condition for jointnleads to the following result:

X

YD 0 WVnCSnsin ̨Sn 1 sinD 0 !VnDH.tan ̨tan/:

Therefore,

IL.Vn/D.tan ̨tan/IL.H /:

Since ̨<andHis negative, then all hangers are in tension. The corresponding

influence line is shown in Fig.3.23.

The influence line for thrustHcan be considered the keyinfluence line, since

thrustHalways appears in any cut-section for the entire structure. This influence

line allows us to calculate thrust for an arbitrary load. After that, the internal force in

any member can be calculated simply by considering all the external loads, the

reactions, and the thrust as an additional external force.

Discussion

For any location of a load the hangers are in tension and all members of the chain are

compressed. The maximum internal force atanyhanger occurs if loadPis placed

at jointC.

To calculate the internal forces in different members caused by an arbitrary fixed

load, the following procedure is recommended:

1.Construct the influence line for the thrust
2.Calculate the thrust caused by a fixed load
3.Calculate the required internal force considering thrust as an additional external
force
This algorithm combines both approaches: the methods of fixed and of moving loads
and so provides a very powerful tool for the analysis of complex structures.