How to Use Partial Derivative Method Due to Axial Strains?

Let's illustrate how to solve the deflections of trusses using the partial derivative method. We shall demonstrate how to solve the rotation of a truss member.

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Let's learn how to use the partial derivative method to solve deflections. This example presents a case of axial strains.

The solution presented is in SI. The author will update the post soon to reflect English units.

To start, we shall consider a simple truss. It has a $20 k N$ point load at $E$ . The value inside the parenthesis () is the member's cross-sectional area $A$ in square centimeters.

Say that we are interested in finding the rotation of member $C E$ . We solve it using the partial derivative method.

For this problem, we'll assume that all members are made of the same material so that the modulus of elasticity $E$ is constant.

Main Solution

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Set-Up Placeholder

Introduce placeholder load representing deflection

The first step is introducing a placeholder couple $M_{P}$ on member $D E$ . It represents the rotation of said member - the thing we want to solve. We apply a pair of equal and parallel forces at the ends of member $D E$ . We'll assume that the rotation of the member is clockwise; hence, the effect of these two forces should be the same.

Formulate S, A, and E Equations

With a placeholder introduced, we formulate axial equations $S$ , $A$ , and $E$ based on the real and placeholder loads.

We get $S$ using the method of joints or sections. Its axial rigidity $A E$ varies per member but not per position $x$ . So, expressing it in terms of $x$ is unnecessary.

The following table shows the summary of each variable per member.

Apply Partial Derivative Equations

At this point, we apply the partial derivative method to solve for member rotation $C E$ .

As the name suggests, we need to find the partial derivative of the axial equation with respect to the placeholder load:

Member BC

$\frac{\partial S}{\partial M_{P}} = \frac{1}{4}$

Member CE

$\frac{\partial S}{\partial M_{P}} = \frac{3}{20}$

Member AD

$\frac{\partial S}{\partial M_{P}} = - \frac{1}{4}$

Member DE

$\frac{\partial S}{\partial M_{P}} = - \frac{1}{4}$

Member BD

$\frac{\partial S}{\partial M_{P}} = 0$

Member CD

$\frac{\partial S}{\partial M_{P}} = 0$

Afterward, we apply the partial derivative equation to solve for the rotation:

$θ_{C E} = \int \frac{S}{A E} \times \frac{\partial S}{\partial M_{P}} d x$

$θ_{C E} = \frac{1}{(20) (E)} \int_{0}^{3} (15 + \frac{1}{4} M_{P}) (\frac{1}{4}) d x + \frac{1}{(25) (E)} \int_{0}^{5} (25 + \frac{3}{20} M_{P}) (\frac{3}{20}) d x$

$+ \frac{1}{(20) (E)} \int_{0}^{3} (- 30 - \frac{1}{4} M_{P}) (- \frac{1}{4}) d x + \frac{1}{(20) (E)} \int_{0}^{3} (- 15 - \frac{1}{4} M_{P}) (- \frac{1}{4}) d x + \frac{1}{(10) (E)} \int_{0}^{5} (25) (0) d x$

$+ \frac{1}{(10) (E)} \int_{0}^{4} (- 20) (0) d x$

Notice that the rotation of $C E$ is in terms of two variables: the placeholder load $M_{P}$ and $x$ . We cannot directly evaluate the integral because of it; however, we recall that the purpose of $M_{P}$ is to represent the required deflection. So, we let $M_{P} = 0$ to have:

$θ_{C E} = \frac{1}{(20) (E)} \int_{0}^{3} (15) (\frac{1}{4}) d x + \frac{1}{(25) (E)} \int_{0}^{5} (25) (\frac{3}{20}) d x + \frac{1}{(20) (E)} \int_{0}^{3} (- 30) (- \frac{1}{4}) d x$

$+ \frac{1}{(20) (E)} \int_{0}^{3} (- 15) (- \frac{1}{4}) d x + \frac{1}{(10) (E)} \int_{0}^{5} (25) (0) d x + \frac{1}{(10) (E)} \int_{0}^{4} (- 20) (0) d x$

$θ_{C E} = \frac{3}{E}$

The positive sign indicates that member $C E$ indeed rotated clockwise from its original position.

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Deriving Partial Derivative Method Due to Axial Strains

This post expands on the general partial derivative work expression derived earlier to consider deflection under axial strains. In the end, we can derive the equations used to compute axial deflections.

Connections

How to Analyze Deflection?

When a structure experiences loads, every structural point will deform and deflect from its original position. This post serves as an introduction to this event.

Created On

June 5, 2023

Updated On

February 23, 2024

Contributors

Edgar Christian Dirige

Founder

References

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