Advanced Methods of Structural Analysis

Problems 259

In this case,
 1 tD10:672 ̨t:

Column 5 is used for calculation of unit displacement of the canonical equation. In
our case,

ı 11 D

24:548

EA

:

These results yield the primary unknown

X 1 D

1t

ı 11

D

10:672 ̨tEA

24:548

D0:435 ̨tEA:

Column 6 contains internal forces in all members of the truss caused by temperature
changes of the lower chord only. These forces, according to (7.22), are equal to
NDNN 1 X 1. Column 7 serves for control of analysis: the sum of all terms of this
column equals to1twith the opposite sign.

Discussion:1. The structure under consideration has one absolutely necessary con-
straint, the reaction of which may be determined from the equilibrium equation. This
constraint is the support bar at the joint 1, which prevents horizontal displacement. If
the truss isexternallystatically indeterminate, then the temperature gradient, which
is related toall membersof the truss, induces a displacement in the direction of
that absolutely necessary constraint, andinternal forces in all members of the truss
induced by temperature gradient are equal to zero.

2. If any member of the truss has been made byunits longer than required,
   then this error of fabrication may be treated as a thermal expansion, i.eD ̨tl,
   wherelis a length of a member; for all other members ̨tlD 0. Canonical equation
   becomesı 11 X 1 C 1 tD0;  1 tD ̨Ntl,whereNis the stress induced in the same
   member by a unit forceX 1.

Problems.......................................................................

Problems 7.1 through 7.6 are to be solved by superposition principle. The flexural
rigidity, EI, is constant for each beam.

7.1.Continuous two-span beam supports the uniformly distributed load q
(Fig.P7.1). Find the reaction of supports and construct the bending moment
diagram.

A

q

BC

ll

Fig. P7.1

Ans:RBD^54 qlI MBD0:125ql^2 I Mmax.0:375l/D0:0703ql^2 :