## Shear Flow In Thin Walled Members

The concept of “shear flow” is very significant in aerospace and this is especially true for thin walled structures. Furthermore we all know thin walled structures dominate aerospace structures.

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Examples of thin walled structures:

- Stiffened fuselage shear panels
- Bulkhead web areas enclosed with stiffeners
- Built up shear web and beam systems with various bays and stiffeners
- Flat or curved panels with edge stiffeners
- Frames with channels and doublers
- Semi-monocoque structures with stringers and longerons etc.

In fact, shear flow is a direct result of vertical shear loads acting in conjunction with bending moments in a beam or built up beam system. What I mean by ‘built up’ is that the structure is an assembly of the web and cap components. In general the beam system is composed of a web and top/bottom caps.

The caps themselves could be multiple parts combined with the web. In the case of fuselage components, there is also the skin that contributes towards the cap areas.

The top and bottom caps are mainly designed to take axial and bending loads, the beam web is predominantly intended to take in plane shear loads. And of course, the fasteners or rivets transfer the resulting shear flow in the system between the components.

But first of all let us talk about some shear flow theory.

## Shear Flow – Bending Stress Theory

To begin with, let us consider a simply supported beam and arbitrary vertical shear and bending moment loads on the left hand side of a segment of this beam. See the FBD of this segment in the figure below:

Firstly, we need to remember that the vertical shear stress on a given area within the beam segment must equal the horizontal plane shear stress for equilibrium. And also the LHS and RHS shears must be equal on segment ‘a’ for equilibrium.

At the left hand side of segment ‘a’:

Bending Stress F_LHS = M*y/I

At the right hand side of segment ‘a’, we now have an additional moment due to V at a distance ‘a’ from the FBD:

Bending Stress F_RHS = M*(y/I)+V*a*(y/I)

We can see from the figure and the equation above, the top hatched segment of the beam segment ‘a’ has an unbalanced force Delta P due to applied bending stress.

The out of balance load on the vertical face of the hatched portion of segment ‘a’ is therefore:

Delta P = V*a*(y/I)*Delta A

This load has to be balanced with the horizontal plane shear load at y’.

The shear stress in the horizontal plane at the top is zero and highest at y’ for this particular portion.

Therefore:

fs*(a*b) = Integral (y’, ymax) (V*a*y/I)dA

Hence:

fs*(a*b) = (V*a*y/I)*Integral (y’, ymax)dA

As we know, Integral (y’, ymax)dA is the first moment of vertical area of the hatched portion, integrated over ‘y’, and it is generally notated as ‘Q’.

Q = Integral (y’, ymax)dA

Hence

fs*(a*b) = (V*a*y/I)*Q

OR

fs = VQ / Ib

Where:

fs = Basic Shear Stress

V = Vertical Shear Force

Q = First Moment of Area A with ‘y’ measured between the overall neutral axis to its centroid

I = Second Moment of Area or Moment of Inertia about beam neutral axis

b = width of the beam

## Shear Flow from Shear Stress

For thin members, the width ‘b’ is the member thickness ‘t’. Therefore the web shear stress:

fs = (VQ / It) PSI

Then the shear flow “q”:

q = fs*t = VQ/I (lb/in)

The equation above represents the shear flow within the thin wall structure members. This shear flow is derived from the applied vertical shear loads as well as bending stresses. The rivet pattern must be able to resist this load without inter rivet buckling or bearing or fastener failures for the beam system to act as one composite member.

Note that shear flow of any rectangular web panel bounded by axial members is constant unless a change in the axial loads in the bounding members causes a shear force, or vice versa. Also note that the above equation is only good for symmetric cross sections.

**Conclusion:**

Thin walled structures experience unique loading such as ‘shear flow’. The types of analysis checks are unique to this kind of a system based on the shear flow. For some more information on theoretical aspects of web shear stress, check this link: **Click Here**

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