As any good student of mechanics knows, we get the best stiffness from a given amount of material by flinging it far away from the neutral axis, which is why I-beams, T-beams, and box beams are so popular. Often, the easiest way to build up these sections (short of extruding or hot-rolling them, which come with material and size limitations) is to join together plates or rectangular sections.
But how can we calculate the required strength for that connection between components of the overall section?
Whether you’re joining pieces of one material or several, shear flow is the force per unit length of your beam required to make the overall section act as if it were one solid piece, giving you full composite action.
Built-Up Sections
Any section made by combining two or more components is known as a built-up section, since it is built up out of various other components.
These components are often rectangular shapes because materials like steel and wood can be had readily in large sheet/plate format, which works well with building large or custom beam shapes.
Shear Flow – The Math
The equation below outlines the shear flow at a particular point on a section, given the applied shear (usually the reaction shear at a support for most load combos controls) and the overall Moment of Inertia of the full built-up section. The units on shear flow, q, are force per unit length.
The one somewhat unfamiliar term in the equation above is Q, the First Moment of Area of the area outboard of the point in question.
Q = the First Moment of Area
Q, the First Moment of Area, is the one semi-unique term in the above equation. Technically it has a nice integral definition, but for all practical purposes here, just draw a line across the section at the joint in question (where your two plates connect in a t-section, for instance) and highlight the area further from the neutral axis of the overall built-up section.
The First Moment of Area for that section is just the area of the highlighted, isolated portion times the distance from the centroid of that isolated portion to the neutral axis of the overall built-up section.
Types of Connection
Connections of the various components into an overall built-up section can be achieved either with a continuous joint, or discrete fasteners at an on-center spacing.
For almost all load cases, the applied shear load, V, isn’t constant but instead varies over the length of the beam. The conservative tactic here is to use the peak shear value over the length of the beam and base all shear flow calculations on this, but this can be over-conservative for some applications. However, in most real applications it is just easier for all involved to use one singular on-center spacing of fasteners for building up sections, as it allows less room for error.
Continuous Connections
Things like glue and continuous weld bead can be used to join components into a built-up section, and here we’re looking for either the strength of a bond for a given width, or the width/weld size required for a given geometry and loading.
If we’re bonding two plates with a structural adhesive of allowable shear stress = 3000 psi, and we have a 300 lbs/in shear flow demand, then we can calculate the following required width, w:
Discrete Connnections
For connectors like nails, screws, rivets, or an on-center weld spacing, we can easily calculate the strength per discrete connection and instead need to find the spacing required to ensure the strength provided exceeds the shear flow demand.
For example, if we were trying to join two pieces of wood to form a built-up T-section, and had a known shear flow at the ends of the beam of 20 lbs/in. Using wood screws that have an allowable shear (the lesser of the screw itself and the bearing of the main and side members) of 100 lbs, then we can calculate:
Other Options for Sizing Connections on Built-Up Sections
The q=VQ/I equation we’ve been focused on in this article does depend on the applied shear force of the section.
There is instead another way to look at shear flow, sizing our connections to ensure that we can achieve a full plastic hinge mechanism and fail in bending. This sort of “full composite action” is used in steel design, and allows for the calculation of the number of shear studs required to join a concrete deck to the steel beams supporting it.
From there, “partial composite action” is typically calculated, which is a more economical design that takes advantage of the ductility of those same shear studs to get much of the result of full composite action with a fraction of the shear studs needed. Expect an article on this in the near future.
Where Else is Shear Flow Used?
Shear flow isn’t just limited to connecting up components of a built-up section. The same q=VQ/I equation is central to the computation of the Shear Center, the point on a section through which loads must be applied to avoid imparting torsion to the shape. For more on Shear Centers, check out my article “What is the Shear Center?”
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