Advanced Methods of Structural Analysis

244 7 The Force Method

deflections for statically indeterminate structure becomes cumbersome. However,
solution of this problem can be significantly simplified, taking into account a fol-
lowing fundamental concept.
Bending moment diagramMPof any statically indeterminate structure can be
considered as a result of application of two types of loads to astatically determinate
structure. They are the given external loads and primary unknowns. It means that a
given statically indeterminate structure may be replaced byany statically determi-
nate structuresubjected to a given load and primary unknowns, which are treated
now asexternalforces. It does not matter which primary system has been used for
final construction of bending moment diagram, since on the basis ofanyprimary
system thefinalbending moment diagram will be the same. Therefore, the unit load
(force, moment, etc.), which correspondsto required displacement (linear, angular,
etc.) should be applied inany statically determinate (!) structure, obtained from a
given structure by elimination ofanyredundant constraints.
This fundamental idea is applicable for arbitrary statically indeterminate struc-
tures. Moreover, this concept may be effectively applied for verification of the
resulting bending moment diagram. Since displacement in the direction of thepri-
mary unknownis zero, then

kD

Z

s

MPMN

EI

dsD

MPMN

EI

D0; (7.12a)

whereMN is the bending moment diagram due to unit primary unknown. This is
called akinematical controlof the resulting bending moment diagram. Equations
(7.12)and(7.12a) are applicable for determination of deflections and kinematical
verification foranyflexural system.
Kinematical verification for structure in Table7.2is shown below. For given
structure the vertical displacement of supportAis zero. We can check this fact
using above theory. Unit state is constructed as follows: the supportAis eliminated
and unit loadPD 1 is applied at pointA. Two bending moment diagrams,MPand
MN,areshowninFig.7.16a. Their multiplication leads to the following results:

verA D

MNPMP

EIi

D

l

6 EI



0  0 C 4 

ql^2

16



l

2



ql^2

8

l



„ ƒ‚ ...

Simpson rule

D0:

Indeed, the vertical displacement of the supportAis zero. Now let us calculate the
slope at supportA. For this we need to show bending moment diagramMPin entire
structure (Fig.7.16a) and apply unit moment at supportAinanystatically determi-
nate structure. Two versions of unit states are shown in Fig.7.16b. Computation of
the slope leads to the following results: