Box Beam Stress Analysis

Aircraft Structural Analysis c S uc u ys s

Chapter 4

Box Beam Stress Analysis

Box Beam Stress Analysis

4.1 Introduction

Š Beam : a slender structural member designed to carry transverse loads

- 3-D, deformable, thin-walled beams

- open or closed section, tapered or none-tapered

Š The transverse loads are transmitted by shear and bending

Š the properly designed cross section

=> the beam can efficiently transmit torsion

4.1 Introduction

Š Fi 4 1 1 ill t t li d i l b f bit ti W

Š Figure 4.1.1 illustrates a cylindrical beam of arbitrary cross section. We will assume for simplicity that the elastic properties of the beam are uniform throughout.

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4.1 Introduction

Š The topics to be reviewed and explored in this chapter are :

„

Area moments of inertia and their transformation properties.

„

The right-hand sign convention, employed throughout this book.

„

The theory of torsion due to Saint-Venant y

„

Bernoulli-Euler flexure theory.

„

Idealized beams (in which the walls carry only shear).

„

Tapered idealized beams. p

4.2 Area Moments of inertia

• Mass Centroid

• 2

moment of Area

P l t f i ti

I

, I

: moment of inertia, I

product of inertia

• Polar moment of inertia

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4.2 Area Moments of inertia

• Rotation

4.2 Area Moments of inertia

Š Table 4.2.1 lists the area moments of inertia for several common shapes.

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4.3 Internal Force Sign Convention g

4.3 Internal Force Sign Convention g

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4.3 Internal Force Sign Convention g

4.3 Internal Force Sign Convention g

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4.4 Saint-Venant Torsion Theory y

Assumption: the angle of twist varies linearly from one end of the rod to the other

Φ is the angle of twist per unit length

Φ Φ

4.4 Saint-Venant Torsion Theory y

Substitution into

C h ’ f l

Cauchy’s formulas, Equation 3.2.8

result

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4.4 Saint-Venant Torsion Theory y

On transverse sections of the torsion bar, the resultant of the , stress distribution must be a pure torque T in the x direction

J : the torsion constant

GJ : the torsion rigidity, a measure of the bar’s resistance to twisting

The y and z components of the resultant force:

4.4 Saint-Venant Torsion Theory y

Š Example 4.4.1

„

Show that the stress distribution in a circular bar in pure torsion is obtained by setting the warping function = const. ψ

„

By Eqn. 4.4.12 ,

„

Since and ,

That is,

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∴

4.4 Saint-Venant Torsion Theory y

For thin-walled circular torsion tubes,

4.4 Saint-Venant Torsion Theory y

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4.4 Saint-Venant Torsion Theory y

4.4 Saint-Venant Torsion Theory y

Example 4.4.2

Š Compare the torsional shear stress and the torsional rigidity of two thin-walled tubes, one of them with a closed section (Figure 4.4.8a) and the other with a small longitudinal gap in the wall (Figure 4.4.8b). g g p ( g )

Š For the closed section, for the open section (cf. Figure 4.4.7e),

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4.5 Bernoulli-Euler Flexure Theory y

When pure bending moments are applied, each point of the axis displaces in direction

perpendicular to the axis,. The cross sections all remain perpendicular to the elastic

curve, and each one undergoes an extremely small rigid-body rotation, with no twist

component. there is no warping : “plane sections remain plane”

4.5 Bernoulli-Euler Flexure Theory y

The position vector of a point in the cross section of beam relative to the beam axis, is yj+zk. The vector displacement of the point consists of the pure translation vj +wk

t ll i t f th ti l th t d t th i id b d t ti

common to all points of the cross section, plus that due to the rigid-body rotation.

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4.5 Bernoulli-Euler Flexure Theory y

The locus of points in a cross section for which ε

=0 is called the neutral axis

4.6 Isotropic Beam Flexure Stress p

The net force is

The resultant is zero since the first area moments and are b t th t id G d th f ∫ ∫

∫ ∫

about the centroid G and therefore zero.

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4.6 Isotropic Beam Flexure Stress p

4.6 Isotropic Beam Flexure Stress p

Example 4.6.1

Figure 4.6.2a shows the first 50 inches of a 200-inch long cantilever beam.

Applied loads act as shown through the centroid G (which in this case

coincides with the shear center) of the free end. Calculate the bending stress ) g distribution on the section at 50 inches from the free end.

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4.6 Isotropic Beam Flexure Stress p

Example 4.6.1

4.7 Thin-Walled Beam Shear Stress

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4.7 Thin-Walled Beam Shear Stress

4.7 Thin-Walled Beam Shear Stress

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4.7 Thin-Walled Beam Shear Stress

Example 4.7.1

Find The shear flow distribution in the thin-walled open section of Figure

4.7.5 if the 1000lb vertical force acts through the shear center. Also, locate

the shear center relative to the vertical web All of the walls have the same

the shear center relative to the vertical web. All of the walls have the same

thickness of 0.1 inch. All dimensions are to the midplanes of the walls.

4.7 Thin-Walled Beam Shear Stress

Example 4.7.1

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4.7 Thin-Walled Beam Shear Stress

Example 4.7.1

4.7 Thin-Walled Beam Shear Stress

Example 4.7.1

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4.7 Thin-Walled Beam Shear Stress

Example 4.7.1

Walls 4 and 5

These walls are identical to their counterparts in the top flange, but the y

coordinate of their centerline is opposite in sign to that of the top flange Therefore coordinate of their centerline is opposite in sign to that of the top flange. Therefore, the shear flows in the bottom flanges will be equal but opposite in direction to

those in walls 1 and 2.

4.7 Thin-Walled Beam Shear Stress

Example 4.7.2

Calculate the maximum shear in the section of Example 4.7.2 if the shear force acts through the vertical web instead of through the shear center, as shown in Figure 4 7 9a

shown in Figure 4.7.9a.

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4.7 Thin-Walled Beam Shear Stress

Example 4.7.2

4.7 Thin-Walled Beam Shear Stress

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4.7 Thin-Walled Beam Shear Stress

Example 4.7.3

Calculate the shear flow in the walls of the box beam with the cross

section shown in Figure 4.7.12 if the webs are effective in bending as well i h

as in shear.

4.7 Thin-Walled Beam Shear Stress

Example 4.7.3

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4.7 Thin-Walled Beam Shear Stress

Example 4.7.3

4.7 Thin-Walled Beam Shear Stress

Example 4.7.3

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4.7 Thin-Walled Beam Shear Stress

Example 4.7.3

4.8 Idealized Thin-Walled Sections

The cross section of an idealized beam consists of thin webs and

concentrated flange areas. The flanges carry all of bending load, while the webs transmit only shear between adjacent flanges.

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Flexure formula from eq 4.6.6

The normal stress σ

is assumed to be uniformly distributes on the

concentrated flange area A g

. The axial load in flange is given by g g y P

= σ

A

If the flange area does not vary with the span, it follows that g y p ,

) (

) ( )

(

dx A d

dx

p dp

f f x

x

= σ

′ = : the flange load gradient

4.8 Idealized Thin-Walled Sections

Š Example 4.8.1

Find the stress in the stringers and the shear flow in the webs at station x= 100 in. of the box beam shown in Figure 4.8.3. The webs are ineffective in bending, and their areas have been lumped at the points indicated in the table in Figure 4.8.3. The skin cross sections are straight lines, except for the curved leading edge, which has an enclosed area table in Figure 4.8.3. The skin cross sections are straight lines, except for the curved leading edge, which has an enclosed area of 35 . in

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4.8 Idealized Thin-Walled Sections

Š Example 4.8.1

Step 1 : locate centroid G of the cross section

Step 2: calculate the moments of inertia about the

axes through the centroid G.

4.8 Idealized Thin-Walled Sections

Š Example 4.8.1

Step 3: find the shear and bending moment at the section 100 inches

Step 4: calculate bending stresses in the stringers.

In this case P = 0, V

= 0, M

=0

) 000 , 100 7

. 173 ( 1 [

2

− ×

σ =

9476

285.7z 1332y

-

6.323)]

- 100,000)(z (37.26

5.758) -

y (

) 000 , 100 7

. 173 ( 26 [ . 37 ) 07 . 83 )(

7 . 173

(

+ +

=

× +

⋅

x

− σ

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9 76 85.7

33 y

4.8 Idealized Thin-Walled Sections

Š Example 4.8.1

Substituting the ith stringer’s coordinates and multiplying by the area gives the stringer loads.

Step 5: Calculate the flange load gradients at x = 100inches

For V

= -1000lb, V

= 0

4.8 Idealized Thin-Walled Sections

Example 4.8.1

Step 6: calculate the shear flow around the section.

Assume all positive shear flows are directed counterclockwise

Step 7: moment equivalence at the section Step 7: moment equivalence at the section

∴

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4.8 Idealized Thin-Walled Sections

Š Example 4.8.2

Find the shear flows in the webs at station x = 1.3 m of

the box beam in Figure 4.8.9.

4.8 Idealized Thin-Walled Sections

Š Example 4.8.2

y

= 0.1208 z

= 0.3578

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4.8 Idealized Thin-Walled Sections

Š Example 4.8.2

4.8 Idealized Thin-Walled Sections

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4.8 Idealized Thin-Walled Sections

Š Method 1

Equally two concentrated areas located at each end of the wall.

The location of the centroid remains the same as the original, but the area moments of

i ti diff t

inertia are different.

4.8 Idealized Thin-Walled Sections

ŠMethod 2

Two stringer had half the wall are and spaced to preserve the centroidal area moment of inertia. f

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4.8 Idealized Thin-Walled Sections

ŠMethod 3

The normal stress on the wall varies linearly from σ

to σ

Let the concentrated areas A1 and A2 at

Let the concentrated areas A1 and A2 at

each end of the idealized wall be selected so

4.8 Idealized Thin-Walled Sections

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4.9 Tapered Idealized Beam Shear Flow p

4.9 Tapered Idealized Beam Shear Flow p

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4.9 Tapered Idealized Beam Shear Flow p

4.9 Tapered Idealized Beam Shear Flow p

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4.9 Tapered Idealized Beam Shear Flow p

E l 4 9 1 Example 4.9.1

Use the average shear flow method to calculate the shear flow in the tapered cantilever beam of Figure 4.9.5. The area of the top and

tapered cantilever beam of Figure 4.9.5. The area of the top and

bottom flanges is 1 in

, while that of the two inner ones is 0.5 in

.

4.9 Tapered Idealized Beam Shear Flow p

Š Example 4.9.1

∴

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4.9 Tapered Idealized Beam Shear Flow p

Example 4 9 2 Example 4.9.2

Use the average shear flow method to calculate the shear flows in the tapered idealized box beam shown in Figure 4.9.7. All cross sections are

i b h i hi h h h h id f h l f

symmetric about the z axis, which passes through the centroid of the left

end of the beam (x=0).

4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.2

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4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.2

∴

4.9 Tapered Idealized Beam Shear Flow p

E l 4 9 2 Example 4.9.2

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4.9 Tapered Idealized Beam Shear Flow p

Example 4 9 2 Example 4.9.2

Use the section-by-section method to calculate the spanwise shear flow variation in the tapered box beam of the previous example.

variation in the tapered box beam of the previous example.

4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.2

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4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.2

4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.2

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4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.2

4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.4

Use the average shear flow method to calculate the shear throughout each of the panels in the idealized tapered box beam pictured in Figure 4 9 13 Table 4 9 4 lists the area and endpoint coordinates relative to the 4.9.13. Table 4.9.4 lists the area and endpoint coordinates relative to the global xyz coordinate system shown, for each of the six stringers.

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4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.4

4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.4

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4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.4

4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.4

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4.9 Tapered Idealized Beam Shear Flow p

Example 4.9.4