Advanced Methods of Structural Analysis

10.1 Construction of Influence Lines by the Force Method 335

Canonical equations (10.13) become

lX 1 C

fl

3

X 2 D

a^2

2

and

l^3

12

X 3 Cl^3 u^2



1

4



u

6



D0:

fl

3

X 1 C

1

5

f^2 lX 2 Dfa^2





1

2

C

2

3

u

1

3

u^2



The solution of these equations leads to the following expressions for the primary
unknowns in terms of dimensionless parameteruDa= l, which defines the location
of the unit forceP:

X 1 Du^2





3

4

C

5

2

u

5

4

u^2



l

X 2 D

15

4

u^2 .1u/^2

l

f

X 3 D 12 u^2





1

4

C

u

6



(10.14)

These formulas should be applied for 0 u0:5.SinceX 1 andX 2 are symmet-
rical unknowns, the expressions for these unknowns for the right part of the arch
.0:5u1/may be obtained from expressions (10.14) by substitutionu! 1 u.
SinceX 3 is antisymmetrical unknown, the sign of expressions forX 3 should be
changed and parameterushould be substituted by 1 u. Influence lines for the
primary unknownsX 1 ,X 2 ,andX 3 may be constructed very easily.
After computation of the primary unknowns we can calculate the reaction and
internal forces at any section of the arch.

10.1.2.3 Reactions of SupportA

The following reactions should be calculated: thrust, vertical reaction, and moment.
Thrust:
HDX 2 D

15

4

u^2 .1u/^2

l

f

for 0 u1:0

This formula presents the thrust of the arch as the function of the dimensionless
parameteru, i.e., this expression is the influence line forH(Fig.10.6c). Maximum
thrust isHmaxD0:2344 .P l=f /and it occurs, when the forcePis located at the
crownC. This formula shows that decreasing of the risef leads to increasing of
the thrustH.
Vertical Reaction:

RADX 3 C^1 D^12 u^2





1

4

C

u

6



C 1 for 0 u0:5