# Understanding Deflections of a Bar

• Howlin
In summary, the conversation discusses the equation for deflection in a bar and its components, including the force per unit length and the second and fourth derivatives of the deflection function, u(x,t). The equation is used to calculate the dynamic response of a beam, with the specific solution depending on the loading and boundary conditions of the bar.
Howlin
Hi

I am looking over deflection in materials and came across the following for a bar u(x,t) and I need some help in understanding it:

F(x,t) = ρA$\frac{∂^{2}u}{∂t^{2}}$ + EI $\frac{∂^{4}u}{∂x^{4}}$

where ρ is the density of the bar, A is the cross-section, F is the force per unit length, E is Youngs modulus and I is the moment of inertia for the cross-section of the bar.

What I can't understand in this is the $\frac{∂^{2}u}{∂t^{2}}$ and $\frac{∂^{4}u}{∂x^{4}}$.

What do they mean and how do you find out what u(x,t) is to work out the 2nd and 4th diritive of it?

From the equations, it appears that the second part of the RHS is the normal Euler-Bernoulli beam equation, where u is the deflection of the beam as a function of position along the length of the beam. In total, this equation appears to be the dynamic form of beam response.

As for u(x,t), that will depend on the loading on the beam.

I agree with SteamKing.

The basic equation of motion for a vibrating bar of uniform cross section is

$$EI\frac{{{\partial ^4}u}}{{\partial {x^4}}} = - \mu \frac{{{\partial ^2}u}}{{\partial {t^2}}}$$

Where μ is the mass of bar per unit length.

This has a general solution

u(x) = Aeax + Be-ax + Csinax + Dcosax

with

$$a = \sqrt[4]{{\frac{{\mu {\omega ^2}}}{{EI}}}}$$

The values of A, B, C & D are determined by the boundary (support) conditions.

## 1. What is the definition of deflection in a bar?

Deflection refers to the displacement or deformation of a bar from its original position when it is subjected to external loads.

## 2. What factors influence the deflection of a bar?

The deflection of a bar is influenced by several factors including the material properties of the bar, its dimensions, and the type and magnitude of the external load applied.

## 3. How is the deflection of a bar calculated?

The deflection of a bar can be calculated using various equations and formulas based on the type of loading and support conditions. These include the Euler-Bernoulli equation, the double integration method, and the virtual work method.

## 4. What are the units of deflection in a bar?

The units of deflection in a bar are typically in inches or millimeters, depending on the unit system used. In SI units, the unit of deflection is meters.

## 5. How can the deflection of a bar be minimized?

The deflection of a bar can be minimized by using a stiffer material, increasing the cross-sectional area of the bar, or by adding additional supports. It can also be reduced by redistributing the load and avoiding concentrated loads on specific points of the bar.

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