Advanced Methods of Structural Analysis

150 6 Deflections of Elastic Structures

Combining (6.2) and data in Table6.1allows us to write the general expressions

for the linear displacementsy.x/and slope.x/for a uniform beam:

EI y.x/DEI y 0 CEI  0 x

X

̇

F.xaF/n

nŠ

; (6.3)

EI .x/DEI  0 

X

̇

F.xaF/n^1

.n1/Š

; (6.4)

whereEIis the flexural rigidity of the beam;Fis any load (concentrated, couple, or

a distributed one);y 0 and 0 are transversal displacement and slope atxD 0 ;aFis

the distance from the origin of the beam to the point of application of a concentrated

force, couple, or to the starting point of the distributed load andnis the parameter,

which depends on the type of the load.

Types of loadFand corresponding parameternare presented in Table6.2.

Ta b l e 6. 2 Initial parameters

method. Parameternfor the

specific loads

Type of loadFFN

Couple M 2

Concentrated load P 3

Uniformly distributed load q 4

The load distributed by triangle law k 5

Equation (6.3) is called the Universal equation of elastic curve of a beam. This

equation gives an easiest way of deriving the equation of elastic curve of uniform

beam and calculating displacement at any specified point. This method is applicable

for a beam with arbitrary boundary conditions, subjected to any types of loads.

Notes:

1.The negative sign before the symbol†corresponds to they-axis directed
downward.
2.Summation is related only to loads, whichare located to the left of the section
x. It means that we have to take into account only those terms, for which the
difference.xa/is positive.
3.Reactions of supports and moment of a clamped support must be taken into ac-
count as well, like any active force.
4.Consideration of all loads including reactions must start at the very left end and
move to the right.
5.Sign of the load factor ̇.F .xaF/n/=nŠcoincides with the sign of bending
moment due to the load, which is located at the left side of the sectionx.
6.Initial parametersy 0 and 0 may be given or be unknown, depending on bound-
ary conditions.
7.Unknown parameters (displacements or forces) are to be determined from the
boundary conditions and conditions at specified points, such as the intermediate
support and/or intermediate hinge.