# Distributed Forces and force density....

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• fog37
In summary, we discussed the difference between concentrated and distributed forces, and how we can find the resultant concentrated force by calculating a surface or line integral of the force density. We also talked about stress and pressure, with stress being a symmetric 2nd-rank tensor and pressure being the isotropic part of stress. We clarified that stress is not a force density, as it is a tensor and not a vector field. We also mentioned that the force density is a different thing, as it is a vector field. Finally, we discussed the possibility of a volume integral of the stress tensor, which represents energy.
fog37
TL;DR Summary
force density or pressure
Hello,
Forces can be concentrated (when acting at a single point) or distributed (when acting over a surface or line).
In the case of distributed forces, we can find the resultant concentrated force by calculating a surface or line integral of the force density ##f(x)## w.r.t. an area or length differential.

This is equivalent to adding together small products of force times infinitesimal areas. Statics books show these types of calculations. For example, $$\int \sigma dA$$
Is the force density ##\sigma =f(x)## to be considered a pressure density or a force density? Pressure is fundamentally normal force per unit area. Is ##\sigma## a pressure density only when the force direction is exactly perpendicular to the surface? The force density may be at an angle at different points on the surface with both a parallel and normal components relative to the surface.

Thanks!

fog37 said:
Summary: force density or pressure

Is the force density σ=f(x) to be considered a pressure density or a force density?
##\sigma## is stress, which is related to pressure but not the same thing. Stress is force density.

Pressure is the isotropic part of stress. Pressure is always normal to a surface, but in general stress can have shear components also.

hutchphd and fog37
fog37 said:
... Statics books show these types of calculations. For example, $$\int \sigma dA$$
Is the force density ##\sigma =f(x)## to be considered a pressure density or a force density? Pressure is fundamentally normal force per unit area. Is ##\sigma## a pressure density only when the force direction is exactly perpendicular to the surface? The force density may be at an angle at different points on the surface with both a parallel and normal components relative to the surface.

Thanks!
Sigma refers only to a normal internal force in the cross-section, which is an indirect effect of the bending load on those beams.
https://en.m.wikipedia.org/wiki/Bending

fog37
Stress is a symmetric 2nd-rank tensor. Let the components be ##\sigma_{jk}=\sigma_{jk}##. It's the force per unit area acting on a surface ##A## of a continuous medium. The total force thus is
$$F_k=\int_A \mathrm{d}^2 A_j \sigma_{jk}.$$
Pressure is a special case. E.g., in an ideal fluid the stress tensor is given (in the local rest frame of the fluid) by
$$\sigma_{jk}=-P \delta_{jk},$$
where ##P## is the pressure.

It's not a force density. This would be a quantity "force per unit volume", e.g., ##\vec{f}=\rho \vec{g}## for the gravitational force close to Earth acting on a fluid.

fog37
vanhees71 said:
Stress is a symmetric 2nd-rank tensor. Let the components be ##\sigma_{jk}=\sigma_{jk}##. It's the force per unit area acting on a surface ##A## of a continuous medium. The total force thus is
$$F_k=\int_A \mathrm{d}^2 A_j \sigma_{jk}.$$
Pressure is a special case. E.g., in an ideal fluid the stress tensor is given (in the local rest frame of the fluid) by
$$\sigma_{jk}=-P \delta_{jk},$$
where ##P## is the pressure.

It's not a force density. This would be a quantity "force per unit volume", e.g., ##\vec{f}=\rho \vec{g}## for the gravitational force close to Earth acting on a fluid.
Thank you. So what I am calling ##\sigma## is the stress tensor (rank 2) whose integral with respect to area is the total force. So far so good. And ##\sigma## is NOT a force density. Why not? Because it is a tensor and not a vector field? The components of the stress tensor are forces either perpendicular or parallel to the surface...

In your last comment about the gravitational force you mention that ##\vec{f}## is instead a force density, force per unit volume, which I get...So the force density is a vector field, a different thing than a rank-2 tensor...Is my understanding correct? To find the overall gravitational force we calculate the volume integral of the force density ##\int f dV##...

To find the overall force on a surface with calculate the surface integral of the stress tensor...

Could we have a volume integral of stress tensor? If so, what does it represent?

vanhees71 said:
It's not a force density. This would be a quantity "force per unit volume"
I disagree. Current density is current per unit area. Not all densities are per unit volume. Stress is force density: force per unit area.

fog37 said:
Could we have a volume integral of stress tensor? If so, what does it represent?
Yes, energy.

Dale said:
I disagree. Current density is current per unit area. Not all densities are per unit volume. Stress is force density: force per unit area.
It's again a matter of definitions. The standard nomenclature in the field-theory literature is that a density of a quantity is per volume and current density of a quantity is per unit area and unit time. In natural units with ##c=1## the dimensions are the same ;-)).

This is an engineering question, not a field-theory question.

## 1. What is a distributed force?

A distributed force is a force that is spread out over a certain area or length, rather than being applied at a single point. It is also known as a force density or distributed load.

## 2. How is force density calculated?

Force density is calculated by dividing the total force by the area or length over which it is distributed. For example, if a force of 100N is distributed over an area of 5m², the force density would be 20N/m².

## 3. What are some examples of distributed forces?

Examples of distributed forces include the weight of an object, wind or water pressure, and the force exerted by a person's body weight on the ground. Other examples include distributed loads on bridges and buildings, and the force applied by a fluid on a surface.

## 4. How do distributed forces affect structures?

Distributed forces can cause stress and strain on structures, which can lead to deformation or failure if the force is too great. Engineers must take into account distributed forces when designing structures to ensure they can withstand the applied loads.

## 5. Can distributed forces be converted to a single point force?

Yes, distributed forces can be converted to a single point force by finding the center of mass or center of pressure of the distributed load. This can be useful in simplifying calculations or analyzing the overall effect of the force on a structure.

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