Vibrations of Structures

Module III: Vibrations of Beams

Lesson 20: Beam Models - II

Contents:

1. The Timoshenko Beam

2. Higher Order Beam Theories

Keywords: Beam models, Shear deformation, Timoshenko beam, Varia-

tional formulation, Higher order beam models

Beam Models - II

1 The Timoshenko Beam

Timoshenko beam model takes into account the shear deformation of a

beam cross-section.

The Newtonian Formulation

The Timoshenko beam requires two field variables: ψ(x, t) for flexure and

ψ

ψ

ψ

w,x

θ

Pure bending

Simple shear

Figure 1: Deformation components of a Timoshenko beam element

θ(x, t) for shear, as shown in Fig. 1. One can write

w,x(x, t) = ψ(x, t) + θ(x, t) (1)

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where w(x, t) represents the displacement field of a neutral fiber of the beam.

The relation (1) allows us to choose of w(x, t) and ψ(x, t) as the two field

variables.

Since the longitudinal strain εx in the beam is produced only from bending,

one can write

εx =(R− z) dψ − R dψ

dx= −zψ,x. (2)

Using Hooke’s law, the longitudinal stress is obtained as σx = −Ezψ,x, and

the bending moment can be expressed as

M = −

∫ h/2

−h/2

σxz dA = EIψ,x. (3)

Similarly, the net shear force acting at a section can be written as

V = GAsθ = GAs(w,x − ψ),

where As = A/κ, A is the area of cross-section of the beam, and κ is known as

the shear correction factor. This factor is introduced to take care of the non-

uniformity in the shear force across the section. For a rectangular section

κ ≈ 1.20, for a circular section κ ≈ 1.11, and for an I-section κ ≈ 2-2.4.

Newton’s second law for the transverse motion yields

(ρA dx)w,tt = V (x+ dx) − V (x) ⇒ ρAw,tt = V,x,

or ρAw,tt = [GAs(w,x − ψ)],x. (4)

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The equation of rotational dynamics can be written as

(ρI dx)ψ,tt = Vdx

2+ (V + dV )

dx

2+ (M + dM) −M

or ρIψ,tt = V +dM

dx,

or ρIψ,tt = GAs(w,x − ψ) + [EIψ,x],x. (5)

The two differential equations (4) and (5) in w(x, t) and ψ(x, t) represent the

dynamics of a Timoshenko beam.

In the case of a uniform beam, we have the simplification

ρAw,tt = GAs(w,xx − ψ,x), (6)

and ρIψ,tt = GAs(w,x − ψ) +EIψ,xx. (7)

Differentiating (7) once with respect to x we have

ρIψ,xtt = GAs(w,xx − ψ,x) + EIψ,xxx. (8)

Solving for ψ,x from (6) and substituting in (8) yields on simplification

ρI

GAsw,tttt + w,tt −

(

I

A+

EI

GAs

)

w,ttxx +EI

ρAw,xxxx = 0. (9)

With the definitions cL =√

E/ρ (longitudinal wave speed), cS =√

G/ρ

(shear wave speed), and rg =√

I/A (radius of gyration), the equation of

transverse motion of a uniform Timoshenko beam can be written as

(

c2L∂2

∂x2−∂2

∂t2

) (

c2Sκ

∂2

∂x2−∂2

∂t2

)

w +c2Sκr2

g

∂2w

∂t2= 0. (10)

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The Variational Formulation

The total kinetic energy density of a beam element consists of the transla-

tional and rotational kinetic energy densities, and is given by

T̂ =1

2ρAw2

,t +1

2ρIψ2

,t. (11)

The potential energy density can be written as

V̂ =1

2EIψ2

,x +1

2GAsθ

2 =1

2EIψ2

,x +1

2GAs(w,x − ψ)2 (12)

Hamilton’s variational principle for the dynamics of the beam yields

δ

∫ t2

t1

∫ l

0

(T̂ − V̂) dx dt = 0,

⇒

∫ t2

t1

∫ l

0

[

ρAw,t δw,t + ρIψ,t δψ,t −

EIψ,x δψ,x −GAs(w,x − ψ)(δw,x − δψ)

]

dx dt = 0. (13)

Integrating by parts and rearranging, we have

∫ l

0

[

ρAw,t δw + ρIψ,t δψ

]∣

∣

∣

∣

t2

t1

dx+

∫ t2

t1

[

− EIψ,x δψ −GAs(w,x − ψ) δw

]∣

∣

∣

∣

l

0

dt

+

∫ t2

t1

∫ l

0

[

(−ρAw,tt + [GAs(w,x − ψ)],x) δw +

(−ρIψ,tt + (EIψ,x),x +GAs(w,x − ψ)) δψ

]

dx dt = 0. (14)

The first integral above vanishes from the statement of the variational princi-

ple. The integrand in the double integral yields the two equations of motion

(4) and (5). A set of possible boundary conditions is obtained from the

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integrand of the second integral in (14) as

[EIψ,x](0, t) = 0 or ψ(0, t) = 0,

and [EIψ,x](l, t) = 0 or ψ(l, t) = 0,

and [GAs(w,x − ψ)](0, t) = 0 or w(0, t) = 0,

and [GAs(w,x − ψ)](l, t) = 0 or w(l, t) = 0.

2 Higher Order Beam Theories

One may extend and generalize the above discussed standard beam models

for improved accuracy (though with higher complexity) as follows. Consider

the expansion of axial and transverse deformations of any material point of

the beam in terms of the transverse coordinate z measured from the middle

plane of the beam as, respectively,

U(x, z, t) = ψ0(x, t) + zψ1(x, t) + z2ψ2(x, t) + . . . (15)

W (x, z, t) = w0(x, t) + zw1(x, t) + z2w2(x, t) + . . . (16)

where ψn(x, t) and wn(x, t) are the field variables. It may be noted that

ψ0(x, t) introduces stretch of the middle plane of the beam. One may then

compute the strain field using the definitions

εxx = U,x, εxz =1

2(U,z +W,x), εzz = W,z,

and the corresponding stress field can be obtained using Hooke’s law.

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From this point, it is convenient to follow the variational formulation to

derive the equations of motion of the beam. The kinetic energy of the beam

can be calculated as

T =1

2

∫ l

0

∫

A

ρ(U2

,t +W 2

,t) dA dx.

The strain energy function may be expressed as

V =1

2

∫ l

0

∫

A

(σxxεxx + 2σxzεxz + σzzεzz) dA dx.

In the above energy expressions, one can carry-out the area integration as

done before. Finally, following the variational procedure, one obtains the

equations of motion for the field variables.

Consider the following two special cases.

(A) ψ1(x, t) = −w,x(x, t) and w0(x, t) = w(x, t):

We have

U(x, z, t) = −zw,x(x, t), and W (x, z, t) = w(x, t).

Thus, εxx = −zw,xx and εxz = εzz = 0, and using Hooke’s law, σxx =

−zEw,xx. The kinetic energy and potential energy expressions are ob-

tained as

T =1

2

∫ l

0

(ρIw2

,xt + ρAw2

,t)dx, V =1

2

∫ l

0

EIw2

,xxdx.

This then leads to the Rayleigh beam theory.

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(a) Euler-Bernoulli theory (b) Timoshenko theory (c) Higher order theories

Figure 2: Visualization of the deformation kinematics in different beam theories (dashed linerepresents deformation under Euler-Bernoulli hypothesis)

(B) ψ1(x, t) = −ψ(x, t) and w0(x, t) = w(x, t):

Here,

U(x, z, t) = −zψ(x, t), and W (x, z, t) = w(x, t).

Hence, εxx = −zψ,x and εxz = 1/2(−ψ + w,x) and εzz = 0. The stress

field is obtained as σxx = −zEψ,x and σxz = 1/2G(w,x−ψ). The kinetic

energy and potential energy expressions are given by

T =1

2

∫ l

0

(ρIψ2

,t + ρAw2

,t)dx, V =1

2

∫ l

0

[EIψ2

,x +GA(w,x − ψ)2]dx.

This yields the Timoshenko beam theory.

The deformation kinematics of Euler-Bernoulli, Timoshenko and higher order

beam theories are visualized in Fig. 2.

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