Advanced Methods of Structural Analysis

10.1 Construction of Influence Lines by the Force Method 325

In case of statically indeterminate truss of the first degree of redundancy, the

expression for influence line of internal force, which is induced at any memberkof

the truss, may be constructed by formula

IL.Sk/DSkIL.X 1 /CIL



Sk^0



; (10.9)

whereSkis the internal force inkth member of a truss caused by the primary

unknownX 1 D 1 in the primary system and IL



Sk^0



is the influence line forkth

member of a truss in the primary system.

Construction of influence lines for internal forces in a statically indeterminate

structure starts from construction of influence lines for primary unknowns IL.X 1 /,

IL.X 2 /;:::.

Construction of influence lines for statically indeterminate structures by the force

method is recommended to perform using the following procedure:

1.To adopt the primary unknowns and primary system. For continuous beams it is
recommended to choose a primary system as a set of simply supported beams.
2.To write the canonical equations of the force method.
3.Compute the coefficientsıikof canonical equations. These unit displacements
present somenumbers.
4.Compute the free termsıiPof canonical equations. These load terms presents
the displacement in the direction ofith primary unknown due to the load which
depends on the location of unit load and therefore presents somefunctions.
5.Solve the canonical equations and construct the influence lines for primary un-
knowns.
6.Construct the influence lines for reactionsand internal forces at specified section.

Below we will show detailed procedure for construction of influence lines by the

force method for continuous beam, hingeless nonuniform parabolic arch, and truss.

As will be shown below, the influence lines for primary unknowns, internal forces,

and reactions for statically indeterminate structures are bounded by the curved lines,

unlike statically determinate structures.

10.1.1 Continuous Beams............................................

Let us demonstrate the above procedure for uniform two-span continuous beamABC

with equal spansl. It is necessary to construct the influence line for internal forces

at a specified sectionk(Fig.10.1a).

Primary System

The structure is the statical indeterminate of first degree. The primary system of

the force method presents two simply supported beams (Fig.10.1b). The primary

unknownX 1 is the bending moment at the middle support.