Shear Flow In Thin Walled Members
The concept of “shear flow” is very significant in aerospace. This is especially true for thin walled structures. Furthermore, we all know thin walled structures dominate aerospace structures.
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 a ‘built up’ system is that the structure is an assembly of the web and cap components. In general, the beam system is composed of a web fastened to the top and bottom cap members or assemblies.
The cap itself can be an assembly of multiple components combined into a cap system. In case of fuselage components, the skin also contributes towards the cap areas.
The top and bottom caps are mainly designed to take axial and bending loads. The web is predominantly intended to take in plane shear loads. The fasteners or rivets transfer the resulting shear flows in the system between the components.
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Firstly, let us discuss some shear flow theory.
Shear Flow – Bending Stress Theory
To begin with, let us consider a cantilever beam. Assume an arbitrary vertical shear load ‘V’ is applied at the right end down. Shear forces and bending moments on the left and right side of a segment ‘dx’ of this beam are shown in the FBD below. Note that the depth of the beam into the page is ‘b’.
Planes A and B define the segment AB of the beam we will study in detail below. In this simple case the moment increases linearly from right to left. Moment at plane B is smaller than moment at plane A. The shear force ‘V’ remains constant from the right end to the fixed left end.
I know there is a lot going on in the image above, let us go through it step by step.
Shear Flow – Bending Stress Theory Figure – Top Row
- The top left view shows the front view of the entire beam. The particular piece or rectangle we are interested in is hatched in black within segment AB. The rectangle is ‘dx’ wide. It also shows the vertical shear force ‘V’ at the tip of the beam. The dashed line through the middle is the beam’s neutral axis (NA) notated as ‘x-x’
- The top right view is a side view showing the thickness or depth of the beam into the page as ‘b’. We see the distance from the beam’s neutral axis x-x to the hatched rectangle’s CG as ‘+y’, below the neutral axis it would be ‘-y’
Shear Flow – Bending Stress Theory – Bottom Row Left and Middle
- The left most view in the bottom row shows the segment AB separated. We can see the applied moment MB at plane B and the slightly larger moment MA at plane A. It shows the balanced vertical shear forces ‘V’ on either side of the segment AB. We also see that MA is larger than MB by the amount “V*dx”, because as we discussed earlier, moment is increasing as we go left
- The middle view in the bottom row shows the bending stress distribution on the left and right faces of the entire segment AB. We see that the peak tensile bending stress is larger at the top fiber on plane A than it is on plane B. This is due to the slightly larger moment MA at plane A than the moment MB at plane B. This trend will continue as we move left towards the fixed end. We also see the equal and opposite compressive bending stresses at the bottom fibers of segment AB
Shear Flow – Bending Stress Theory – Bottom Row Right
- Moving on to the right most view mage in the bottom row, we now get into the nitty gritty. The little hatched rectangle (front area of the cube) is separated out. Top top cube above the hatched rectangle shows the depth ‘b’ for reference, and width ‘dx’. To its right we see the theoretical total applied stress sigma_dA_B on the right face of the cube. Similarly the theoretical total applied stress sigma_dA_A acts on the left face (hidden) of this cube.
- Below the top cube, we see the hatched area of interest. The stress, and hence the load (P), acting on these faces varies linearly as shown by the blue trapezoids. Loads PA and PB are the total forces acting at the CG of this linearly varying stress distribution. As noted earlier, since MA is slightly larger than MB, so is the stress. Therefore, the load PA is slightly larger than load PB. You think of this imbalance being reacted by the shear force that acts on the bottom face of the cube or hatched rectangle. This force (PdA_S) is indicated with a half arrow pointing to the right under the hatched rectangle. It is important to note that at the top surface, shear stress is zero as it is a free face. The Half force couple on the left and right of the hatched rectangle balances the moment cause by PdA_S about the top free edge. These half arrow shear flows (we will discuss this in more detail below) always point arrow to arrow or tail to tail. So if one direction on one edge is known, there rest automatically fall into place.
- The bottom cube in the right view shows the equations to calculate the loads on the left and right faces, PdA_A and PdA_B respectively. It is basically Mc/I, but integrated over dA for each small dA segment, with a varying ‘y’ in place of ‘c’.
Shear Flow – Derivations
Let us begin with a static equilibrium equation with right being positive:
Therefore:
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 overall neutral axis
b = Depth or thickness of the beam into the page
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 member. The above equation is only good for symmetric cross sections. 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. In a separate blog post, we can get more in depth into how the shear flow increases towards the neutral axis as “Q” increases and then goes back to zero below the neutral axis towards the other end.
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
So that is all for now folks, what did you think about this post? Comment below the post….
Cheers…
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