ENG5312 – Mechanics of Solids II
47
Deflection of Beams and Shafts
Elastic Curve

The elastic curve is the deflection diagram of the longitudinal axis that
passes through the centroid of each cross-section of a beam (or shaft).

Supports restrict the deflection of a beam:
i.
Supports that resist force (e.g. pin joint) restrict displacement
only; and
ii.
Supports that resist a moment (e.g. fixed) restrict displacement
and slope.

The elastic curve may be drawn from intuition:

Or the elastic curve may be constructed from the variation of bending
moment obtained from a bending moment diagram.

Sign convention: positive internal moment bends a beam concave up,
and a negative internal moment bends a beam concave down.
ENG5312 – Mechanics of Solids II

When the bending moment changes sign there will be an inflection point
(i.e. change from concave up to concave down, or vice versa).
48
ENG5312 – Mechanics of Solids II
49
Moment-Curvature Relationship



Define the x -axis along the straight longitudinal axis of the beam and the
v -axis vertically up (where v is used to measure the vertical displacement
of the centroid).

Define
 a local y -co-ordinate used to measure position up from he neutral
axis.

Consider the beam to be initially straight and to be deformed elastically by
loadsapplied perpendicular to the x -axis, in the x,v plane. The
deformation will be caused by bending moment and shear force. If the
beam length is much greater than its depth, then the greatest deformation
will be caused by bending.

The bending moment will cause the beam to deform such that an angle
d is created between the cross-sections at x and x  dx .

Define the radius of curvature,  , as the distance from the neutral axis to
the center of curvature, O' .






ENG5312 – Mechanics of Solids II
50

Any arc in the element (other than the neutral axis) will experience normal
strain.

For the segment ds:

ds'ds
ds

 But , ds  dx  d (i.e. dx does not change) and ds' (  y)d .

So  can be written as: 



  y d d   y
d


Or
1




y
1 M
   EI
(47)
Or
1



Assuming the material is homogeneous and behaves in a linear-elastic
manner, Hooke’s Law applies, therefore,    / E and since the flexure
/ I :
formula applies,   My






Ey
(48)
Where

o
 is the radius of curvature at a specific location;
o
M is the internal bending moment;

o
I is the moment of inertia about the neutral axis; and

o
EI is the flexural rigidity (note if EI ,   (i.e. less curvature or


bending…thus the invention of the I-beam).


ENG5312 – Mechanics of Solids II

51
Note: The sign of  depends on the sign of M .
i.
M  0 , concave up,   0, i.e center of curvature is above the
beam; and


ii.
M  0 , concave down,   0, i.e. center of curvature below the

beam.



Slope and Deflection by Integration

The goal is to obtain an equation for the deflection of a beam as a function
of x , i.e. v  f (x).

From calculus:


1


d 2 v / dx 2
1 dv / dx 
2 3/ 2

M
EI
(49)

This equation is the elastica, a second-order non-linear differential
equation. It will give the exact elastic curve for bending only.
Unfortunately,
 only a few solutions are available.

Usually, the desired deflections are small, therefore, the elastic curve is
shallow, and the slope dv / dx must be small, so (dv / dx) 2 1 and may
be neglected in Eq. (47) (for small deflections). Then:
d 2v M

dx 2 EI



(50)
Other forms of this equation may be derived using: V  dM / dx and
w  dV / dx




d  d 2v 
EI
 V (x)
dx  dx 2  
(51)
d 2  d 2v 
EI
 w(x)
dx 2  dx 2 
(52)
ENG5312 – Mechanics of Solids II

52
If the flexural rigidity is constant (realistic for many beams):
EI
d 4v
 w(x)
dx 4
d 3v
EI 3  V (x)
dx

EI

d 2v
 M (x)
dx 2
(53)
(54)
(55)

Either of these equations may be integrated to obtain v(x) . Usually, Eq.
(55) is used, i.e. determine M (x) , integrate twice, and determine the two

constants of integration.

Sign convention: Positive sign convention isshown in the two figures from

Hibbeler below. Remember,
v is measured positive up, and since small
deflections have been assumed,   tan   dv / dx .


ENG5312 – Mechanics of Solids II
53

Note: In general, the loading on a beam is discontinuous, and several
functions must be written for the internal moment, with each function being
valid in specific locations. Also, each function for the moment may use a
different co-ordinate axis. So…several functions may be required to
specify the deflection of a beam, with each function being valid in specific
regions of the beam. At the intersection of two functions, continuity must
be maintained (i.e. v and dv / dx ).

Boundary Conditions:
o Typical
conditions used to evaluate the constants of

 boundary
integration are shown in Table 12-1 of Hibbeler:

Continuity Conditions:
o Used to guarantee a continuous elastic curve when more than one
x -co-ordinate is required to specify the bending moment ( and
therefore, the elastic curve) for the beam. Both the slope and
displacement must be matched.
