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__Beam Deflection__

** Introduction**

A beam with a straight
longitudinal axis subjected to transversal loads acting in its longitudinal
plane of symmetry deforms and the resulting curve is called the **deflection curve**. In the previous
lecture, the curvature of the deflection curve was used to determine the stress
and strain distribution in the cross-section of a beam under the restrictions
established for pure and nonuniform bending. In this lecture, the equation of
the defection curve will be derived and consequently, the displacements at any
point of the beam may be calculated. The calculation of beam deflections is an
important part of structural analysis and design. The deflections are limited
to prescribed tolerances imposed by the functionality of the particular
structural element.

The cantilever beam pictured in Figure 1 deforms in its vertical plane of symmetry under the action of the exterior transversal loading.

Figure 1 Example of Beam Deflection

The deflection curve and its slope
are mathematically represented by the real functions_{}and_{},
respectively. The vertical distance measured from a given point located on the
undeformed axis of the beam to the corresponding point located on the
deflection curve is called the **transverse
displacement**.

** Qualitative
Interpretation of the Deflection Curve**

For the general case of
nonuniform bending the equation relating the bending moment _{} with the radius of curvature _{}is:

_{} (1)

where _{} - modulus of elasticity;

_{}-
moment of inertia about the horizontal centroidal axis of the cross-section;

_{}-
cross-section bending moment;

_{}-
radius of curvature of the deflection
curve.

** Note:** If the material is homogeneous linear elastic and the beam is of
uniform cross-section the product

From the relation given by
equation (1), the deflection _{} can be anticipated using the sign convention established
in Lecture 7 and re-plotted in Figure 2. *The
moment diagram is always plotted at the beam side where the fibers are subjected
to tension, while the curvature center is placed on the opposite side.* When
the bending moment is positive the deflection curvature is concave, while for
the negative bending moment the deflection curvature is convex. At the supports
the deflection must correspond to the prescribed support constraint condition. Locations
where the deflection curve changes from concave to convex, or vice versa, are
called **inflection points**.

Figure 2 Sign Convention for Beams in Bending

An example of qualitative interpretation of the deflection curve is shown in Figure 3. The moment diagram changes from positive to negative in the interval AB at point D. In between points A and D the moment is positive and the curvature center is located above the deflection curve. Between points D and B the moment diagram is negative and, thus, the curvature center is located below the deflection curve. The change in curvature takes place at point D, which is an inflection point. On the overhanging end BC, the moment is negative and the curvature is convex. Considering that at the supporting points the beam is constrained not to move vertically, the qualitative deflection diagram, as shown in Figure 3, can be sketched.

Figure 3 Example of Qualitative Interpretation of Deflection Curve

** Differential
Equations of the Deflection Curve**

From Calculus it is known that the
slope of a real function at any particular point of its defined continuity
interval is the first derivative of the function. For the case in point, the
real function is represented by the beam displacement curve_{}.
Using the notation shown in Figure 1 the following relation is written:

_{} (2)

*Under the assumption of small displacements,* the first derivative
is a very small value:

_{} (3)

consequently, the angle can approximate by its tangent:

_{} (4)

In Calculus the relation between the radius of curvature_{}and
the deflection _{}is
established as:

_{} (5)

Under the small displacement assumption (3), equation (5) can be simplified and written as:

_{} (6)

Substituting equation (6) into equation (1) yields the following equation:

_{} (7)

The **moment-curvature equation**, a second order differential equation, is
obtained from equation (7) and expressed in the following standard form:

_{} (8)

where the notation _{} is employed.

Consider the differential
relation between the transverse loading and the cross-section stress resultants,
_{}and
_{},
previously obtained from the equilibrium of the beam infinitesimal volume
element:

_{} (9)

_{} (10)

where _{}and
_{}are
the vertical shear force and the bending moment, respectively.

Differentiating the
moment-curvature equation (8) and using equations (9) and (10) the **shear-deflection** and** load-deflection** equations are
obtained:

_{} (11)

_{} (12)

The load-deflection equation (12)
is a *forth-order differential equation*.

If the beam is made from a homogeneous linear elastic material and has a constant cross-section in the continuity interval of the bending moment or transversal loading equations (8) and (12) became:

_{} (13)

_{} (14)

The general theory of
differential equations with constant coefficients can be employed to obtain the
solution of the second-order differential equation (8) and forth-order
differential equation (12).* The
continuity intervals pertinent to the functions involved in these differential
equations, the bending moment and transverse load, must be recognized in order
for the integration process to be properly conducted.*

** ****Integration
of the Moment- Curvature Differential Equation**

Integration of the second-order
differential equation (8) on an interval of continuity for the bending moment_{}
yields the following relations:

_{} (13)

_{} (14)

For the case of a beam with constant
bending stiffness _{},
the integrals expressed in equations (13) and (14) may be simplified as
follows:

_{} (15)

_{} (16)

The integration constants _{} and _{} are calculated by imposing the **boundary conditions **of the specific
problem at hand.

** ****Integration
of the Load-Deflection Differential Equation**

Successive integration of the forth-order differential equation (12) in a continuity interval of the transverse loading function yields the following relations:

_{} (17)

_{} (18)

_{} (19)

_{} (20)

The integration constants_{},_{},
_{} and _{} are calculated by imposing the boundary
conditions applicable to the specific problem.

__Note:__ *The moment-curvature
equation (8) can be used only if the bending moment variation is known, the
case for statically determinate structures. The load-deflection curve equation
(14) requires only that the variation of transverse load be known and,
consequently, can be used for either statically determinate or indeterminate
beams.*

For the case of a beam with constant
bending stiffness _{},
the integrals expressed in equations (17) through (20) may be considerably simplified
as follows:

** **_{} (21)

_{} (22)

_{} (23)

_{} (24)

** ****Boundary and
Continuity Integration Conditions**

As previously stated, the functions (15) though (21) obtained above, represent the general solutions. Only after the boundary conditions are imposed and the integration constants determined the solutions became representative for a specific case study.

The boundary conditions commonly encountered in the application of the equations (15) and (16) or equations (21) through (24) are presented in Table 1.

In general, the transverse load _{}and
the bending moment _{}functions
are described for a particular case by a number of continuity intervals.
Therefore, the **continuity conditions**
at the common ends of the intervals must be described in order for the
constants to be calculated. Each continuity interval is treated as an
independent interval and the boundary and continuity conditions are applied.
The most common continuity conditions are summarized in Table 2.

__Table 1 Boundary Conditions__

__Table 2 Continuity Conditions__

** Examples**

The methodology used in the application and integration of the moment-curvature (8) and load-deflection (14) differential equations is illustrated in examples 4.1 and 4.2, respectively.

** ****Application
of the Moment-Curvature Equation**

The deflection curve is required
for a cantilever beam subjected to a concentrated force _{} and a concentrated bending moment _{} both acting at the tip of the beam, point B. The
beam is characterized by a constant cross-section along its entire length, with
geometry and loading as shown in Figure 4.

_{}constant (25)

The corresponding reactions, _{} and _{},
are found using the equilibrium equations:

_{} (26)

_{} (27)

Figure 4 Cantilever Beam

The moment-curvature equation (8)
requires knowledge of the bending moment function and identification of the
continuity intervals. For the case in point, the bending moment _{}diagram
is continuous on the entire length of the beam and is expressed as:

_{} (28)

Using equation (28) in the moment-curvature equation (13), the problem specific differential equation is obtained:

_{} (29)

Integrating the differential equation (29) twice yields the following expressions for deflection and slope:

_{} (30)

_{} (31)

The integration constants _{}and
_{}are
identified using the boundary conditions at point A:

_{} _{} (32)

_{} (33)

Solving the algebraic equations (32)
and (33) by substitution of equations (30) and (31) the integration constants _{}and
_{}are
found as:

_{} (34)

_{} (35)

Substituting equations (34) and (35) into equations (30) and (31), the final deflection and slope expressions are obtained:

_{} (36)

_{} (37)

The maximum deflection value is obtained at the tip of the cantilever:

_{} (38)

and the corresponding rotation

_{} (39)

The deflection curve is a cubic (third-order) polynomial and is schematically plotted in Figure 5

Figure 5 Deflection curve

Equations (38) and (39) may be written
for the case when only the concentrated force _{} is considered:

_{} (40)

_{} (41)

In the absence of the
concentrated force _{} equations (38) and (39) become:

_{} (42)

_{} (43)

** ****Application
of the Load-Deflection Equation**

The simply supported beam shown
in Figure 6 is subjected to a concentrated vertical force _{}acting
at distance _{} from the fixed support A. The beam has a
constant flexural rigidity along its entire length.

_{}constant (44)

The reaction force and moment at point A are obtained by solution of the equilibrium equations:

_{} (45)

_{} (46)

Figure 6 Cantilever Beam

The transverse load has two continuity intervals, AB and BC, and, consequently, the forth-order differential equation must be integrated for each one of them as:

- For interval AB (the interval origin is point A) the transversal load is zero:

_{} (47)

Using equation (47) in equations (21) through (24) yields the following:

_{} (48)

_{} (49)

_{} (50)

_{} (51)

- For interval BC (the interval origin is point B) the transversal load is also zero:

_{} (52)

Equations (21) through (24) are written considering equation (52) as:

_{} (53)

_{} (54)

_{} (55)

_{} (56)

Eight (8) integration constants must be calculated. Four (4) boundary conditions (at points A and C) and four (4) continuity conditions (at point B) are employed. The boundary conditions are expressed as:

- Point A
_{}x=0 (interval AB)

_{} _{} (57)

_{} _{} (58)

- Point
C
_{}x=b (interval BC)

_{} _{} (59)

_{} _{} (60)

- The continuity conditions at point B (both intervals) are:

_{}

_{} (61)

_{}

_{} (62)

_{} _{} (63)

_{} _{} (64)

Solving the system of algebraic equations (57) through (64) the integration constants are calculated:

_{} _{} _{} _{} (65)

_{} _{} _{} _{} (66)

Substituting the integration constants (65) and (66) into equations (48) through (51) and (53) through (56), respectively, the variation of the shear force, bending moment, slope and deflection for each interval are obtained:

- For the interval AB (the interval origin is point A)

_{} (67)

_{} (68)

_{} (69)

_{} (70)

- For the interval BC (the interval origin is point B)

_{} (71)

_{} (72)

_{} (73)

_{} (74)

The deflection and slope at points B and C are:

_{} (75)

_{} (76)

_{} (77)

_{} (78)

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