Euler-Bernoulli Beam Theory

A bending test is an experimental set-up to determine structural deformation behaviour by applying predefined forces to a specimen, see Fig. 2.1. For a bending test, a specimen is shaped as a slender beam with specific geometry.

Then the beam specimen is clamped with bearings. At specific positions on top of the beam, forces of predefined strength are applied. For a given bending test configuration, the deflection of the beam in the deformed state can be computed by the Euler-Bernoulli beam equation (cf. Gross, Hauger, Schröder, Wall, and Bonet 2011, pp. 125–129). To this, observe the local balance of angular momentum in Eq. (2.96) which satisfies the balance of mass in Eq. (2.50)

% d

dt θ_krω_r+s_k

= ∂

∂x_j ε_klix_lσ_ij +m_kj

+% ε_klix_lf_i +l_k

. (2.163)

For materials with non-complex inner structure, the spins_kas well as the influences of the surface and volume couple, m_kj andl_k, are neglected, thus we have a symmetric stress tensor in Eq. (2.105). For statically bended beams, the dynamic part in the left-hand side of the balance equation vanishes. Furthermore, the gravitational specific volume forcef_i can be ignored due to the much greater applied bending forces. Consequently, we obtain the momentum equilibrium

0 = ∂

∂x_j ε_klix_lσ_ij

. (2.164)

FIELD EQUATIONS|BASICS OF CONTINUUM MECHANICS 33

F⁰

F¹

F²

F³

x₁

˘ x^p

x^p

Figure 2.2: The Method of Sections generates afree-body diagramfor revealing the inner stresses within a beam. The forcesF^pare applied on the rectangular traction areas with depthdand widthwwhich are marked in red.

Integrating over the volume of the beam body, followed by the application of the Gauss’ theorem, we find 0 =

ˆ ∂

∂x_j ε_klix_lσ_ij dV =

ˆ

ε_klix_lσ_ijn_jdA=

= ˆ

∂B_internal

ε_klix_lσ_ijn_jdA+ ˆ

∂B_external

ε_klix_lt_idA=M_k^internal+M_k^external. (2.165)

In the last step, we decompose the total momentum into internal and external parts in order to reveal the inte-rior stress of the beam. This is possible by means of themethod of sections, see for example in Gross, Hauger, Schröder, Wall, and Rajapakse (2009, p. 10) or in Gere and Goodno (2013, p. 8). As shown in Fig. 2.2, we section a imaginary cut to the beam perpendicular to direction1at positionx₁and examine the right-hand side part. The internal momentumM_k^internalis computed from the cutting area at the cutting pointx₁. In a bending testing apparatus, the beam specimen is subjected to two types of external forcesF^p: active and reactive. Predefined line forces, applied on top of the beam at specific positionsx^p, are considered active, while the reactive forces oc-curs as a response to the constraints induced by the supports at positionsx^p. The superscriptpidentifies uniquely the active and reactive forces and are numbered from0toN. A line force comprises the acting forceF^pthat are distributed on rectangular area of depthdand widthwat positionx^p. The rectangular areas are marked in red in Fig. 2.2. The external momentumM_k^externalis determined by the external forcesF^pon the right-hand side part at the cutting pointx₁.

To proceed, we apply the so-calledsemi-inverse methodfor M_k^internal=

ˆ

∂B_internal

ε_klix_lσ_ijn_jdA . (2.166)

In this method, the displacement functions are supposed to have certain form by geometrical considerations and simplified assumptions. For the Euler-Bernoulli beam theory, we have to ensure that the tested beam specimen fulfils the Bernoulli’s hypothesis that the cross sections of the beam remain plane and perpendicular in the bent

α

α

u₁ = tan α x₂

x₁ x₂

x₂ x₁

Figure 2.3: Explanation of the semi-inverse method.

configuration to the set of lines that don’t extend or contract. This can be ascertained for small deformation. The black dashed line in Fig. 2.1 indicates this set of lines that is known as theneutral surface. Furthermore, in Fig. 2.1, the red line shows that the cross section remains perpendicular in the deformed beam. This in turn leads to the following displacement functions

u₁ = tan α

x₂= ∂u

∂x₁x₂, u₂ =u= ˘u x₁

, u₃ = 0. (2.167)

The displacementu₃vanishes due to the set-up of the bending apparatus that excludes deformation in direction3. For small deformation, the displacementu₂depends only onx₁. The displacementu₁is determined by the cross section in the bent configuration of the beam, see Fig. 2.3. Whileu₁is a function ofx₁andx₂, its slopetan α depends only onx₁due to the Bernoulli’s hypothesis. The slopetan α

is determined as depicted in Fig. 2.4.

It shows that the slopetan α

is also determined by an infinitesimal change of displacementu₂,∂u₂, divided by an infinitesimal change of positionx₁,∂x₁.

For small deformation, we use the linear strain in Eq. (2.28) with the displacement functions to determine non-zero strains as follows

ε₁₁= ∂u₁

∂x₁ = ∂²u

∂x₁²x₂andε₁₂= 1 2

Å∂u₁

∂x₂ +∂u₂

∂x₁ ã

= ∂u

∂x₁. (2.168)

Since the total length of a slender beam is much greater than the deflection for small deformation, it is feasible to assume that the inclination of the deformed beam is negligibly small

∂u

∂x₁ ≈0. (2.169)

Furthermore, the beam consists of isotropic material which obeys Hooke’s law. This results in the following non-zero entries of the stress tensor

σ₁₁ = λ+ 2µ

ε₁₁, σ₂₂ =σ₃₃=λε₁₁. (2.170)

The Poisson’s ratioνvanishes as a consequence of the chosen displacement functions ν=−ε₂₂

ε₁₁ =−ε₃₃

ε₁₁ = 0, (2.171)

therefore the Lamé constants become

λ= 0, µ= E

2 . (2.172)

The only non-vanishing stress entry left is

σ₁₁ =Eε₁₁. (2.173)

Therefore, the remaining internal momentums are M₂^internal=

ˆ

ε₂₃₁x₃σ₁₁n₁dA= ˆ

x₃σ₁₁dx₂dx₃= ˆ

x₃Eε₁₁dx₂dx₃=

FIELD EQUATIONS|BASICS OF CONTINUUM MECHANICS 35

∂x₁

∂u₂

α

α

Figure 2.4: Explanation of the semi-inverse method.

= ˆ

x₃Ex₂ ∂²u

∂x₁² dx₂dx₃ =E ∂²u

∂x₁²

h

ˆ2

−^h₂

x₂dx₂

w

ˆ2

−^w

2

x₃dx₃ =

=E ∂²u

∂x₁² 1 2

x₂^{2 h}₂

−^h₂

| {z }

=0

1 2

x₃^{2 w}₂

−^w₂

| {z }

=0

= 0 (2.174)

and

M₃^internal= ˆ

ε₃₂₁x₂σ₁₁n₁dA= ˆ

−x₂σ₁₁ −1

dx₂dx₃= ˆ

x₂Eε₁₁dx₂dx₃=

= ˆ

x₂Ex₂ ∂²u

∂x₁²dx₂dx₃ =E ∂²u

∂x₁²

w

ˆ2

−^w

2 h

ˆ2

−^h₂

x₂²dx₂dx₃

| {z }

=I₂₂

=EI₂₂ ∂²u

∂x₁² ,

whereI₂₂is known asmoment of inertia of plane area. For the external momentum

M_k^external= ˆ

∂B_external

ε_klix_lt_idA , (2.175)

the tractionst_i that are acting on the beam are considered. Due to the apparatus set-up, the active and reactive forces point along the directionx₂-axis. Therefore, the only remaining traction vector component is

t₂ = ˘t x₁, x₂

=







t^p forx₁ ∈

x^p−^d₂, x^p+^d₂

, x₂ = ^h₂, t^p forx₁ ∈

x^p−^d₂, x^p+^d₂

, x₂ =−^h₂, 0 otherwise,

(2.176)

wheret^pis a constant for each acting force pointp. The remaining external momentums are M₁^external =

ˆ

ε₁₃₂x₃t₂dA= ˆ

−x₃t₂dA= ˆ

ε₁₃₂x₃t₂dA= ˆ

−x₃t₂dx₁dx₃=

=

N

X

p=1

w

ˆ2

−^w

2

xˆ^p+^d₂

x^p−^d₂

−x₃t^pdx₁dx₃ =

N

X

p=1

−1 2

x₃^{2 w}₂

−^w

2

| {z }

=0

t^pd= 0. (2.177)

and

M₃^external= ˆ

ε₃₁₂x₁t₂dA= ˆ

x₁t₂dA= ˆ

x₁t₂dx₁dx₃=

=

N

X

p=1

w

ˆ2

−^w

2

xˆ^p+^d₂

x^p−^d₂

x₁t^pdx₁dx₃=

N

X

p=1

x^pt^pdw

| {z }

=F^p

=

N

X

p=1

x^pF^p. (2.178)

We obtain the momentum equilibrium for the Euler-Bernoulli beam as follows

0 =M₃^internal+M₃^external=EI₂₂ ∂²u

∂x₁² +

N

X

p=1

x^pF^p. (2.179)

Fork = 1andk = 2momentum equilibrium are fulfilled without contradiction. The bending momentMin dependence of positionx₁is determined as resistive moment of the external momentumM₃^externalat sectioning pointx₁. Each moment lever armx^pis computed by the distance between the imaginary cut positionx₁and the force acting pointx˘_p

x^p =x₁−x˘^p. (2.180)

The bending moment is computed as follows

M₃^external=

N

X

p=1

x^pF^p =

N

X

p=1

x₁−x˘^p

F^p = ˘M x₁

=M . (2.181)

We rewrite the momentum equilibrium in the form known as the Euler-Bernoulli beam equation EI₂₂ ∂²u

∂x₁² =−M . (2.182)

Im Dokument The measurement- and model-based structural analysis for damage detection (Seite 44-48)