Advanced Methods of Structural Analysis

7.3 Analysis of Statically Indeterminate Structures 233

Final diagram for axial forceNis presented in Fig.7.11j. Since the memberDBhas

no constraints for its horizontal displacement, a normal force in this member is zero.

The similar conclusion related to the memberDC.

6.Reactions of supports.Internal force diagramsM,Q,andNallow us to show all

reactions of the support. The negative shear 3.756 means that horizontal reaction

at supportCequals 3.756 kN and directed from right to left. The positive shear

6.444 means that horizontal reaction at clamped supportAequals 6.444 kN and

this direction of the reactionRAproduces the positive shear, etc. All reactions

of supports are shown in Fig.7.11k.

Now we can perform controls of the obtained reactions of supports. Equilibrium

equations for the frame in whole are

X

XDq 5 6:2443:756D 10  10 D 0

X

YDPC6:108C1:892D 8 C 8 D 0

X

M 7 D 2  5 2:5C8:0366:108 4 6:244 5 C3:756 3 C1:892 6

D55:65655:652Š 0

Therefore, equilibrium of the structure in a whole is held. It is left as an exercise for

the reader to check the equilibrium of some parts of the system (for example, use

a cut section through left/or right at point7) and considering the equilibrium of the

either part of the structure.

Discussion:

1.For any statically indeterminate structure subjected to action of arbitrary exter-
nal load, the distribution of internal forces (bending moment, shear and normal
forces), as well as reactions of supports depend only on relative stiffnesses of the
elements, and does not depend on their absolute value of flexural stiffness,EI.
2.Two free-body diagrams for jointDtakes into account different internal forces:
the Fig.7.11g contains only the bending moments, while Fig.7.11i contains the
shear and axial forces. It happens because the section is passedinfinitely closeto
jointD.
3.The unit bending moment diagrams are used at different steps of analysis, so the
effectiveness of these diagrams is very high.

7.3.3 Analysis of Statically Indeterminate Trusses

Statically indeterminate trusses are geometrically unchangeable structures for which

all reactions and all internal forces cannotbe determined using only the equilibrium

of equations. Figure7.12presents three types of statically indeterminate trusses.

They are (a) externally, (b) internally, and (c) mixed statically indeterminate trusses.