# Derivation of sagging expression

• Alpharup
In summary, the formula for the sag of a bar under load is δ = W l^3/(4bd^3Y), where Y is the young's modulus of the wire. This can be derived using calculus and is explained in the Wikipedia article "Euler–Bernoulli beam theory". The article also discusses the three-point bending method and the behavior of shear and bending moment in a loaded beam.
Alpharup
A bar of length l, breadth b, and depth d when loaded at the centre by a load W sags by
an amount given by
δ = W l^3/(4bd^3Y). Here Y is the young's modulus of the wire. My textbook says that we have to use little calculus to derive this expression. Can you please derive this?

mohamed hassan
It gets pretty complicated. But take a look at the Wikipedia article "Euler–Bernoulli beam theory", especially the section "Three-point bending" and see if that is any help.

"The shear is constant in absolute value: it is half the central load, P/2. It changes sign in the middle of the beam. The bending moment varies linearly from one end, where it is 0, and the center where its absolute value is PL/4. The deformation of the beam is described by a polynomial of third degree over a half beam (the other half being symmetrical)."

## What is "Derivation of sagging expression"?

"Derivation of sagging expression" refers to the scientific study of the physical changes that occur in the face as people age, specifically the drooping or sagging of facial features such as the cheeks, eyelids, and jawline.

## What causes sagging expression?

Sagging expression is primarily caused by the loss of collagen and elastin in the skin as we age. These proteins are responsible for maintaining the skin's elasticity and firmness. Other factors that contribute to sagging expression include genetics, lifestyle choices, and environmental factors.

## Can sagging expression be prevented?

While sagging expression is a natural part of the aging process, there are steps that can be taken to slow it down. These include protecting the skin from sun damage, maintaining a healthy lifestyle, and using skincare products that promote collagen production.

## Are there any treatments for sagging expression?

Yes, there are various treatments available for sagging expression, including cosmetic procedures like facelifts and dermal fillers. There are also non-invasive options such as radiofrequency therapy and laser treatments that can help tighten and lift sagging skin.

## At what age does sagging expression typically occur?

Sagging expression can begin to occur in your 30s or 40s, but it is more noticeable in your 50s and beyond. However, the exact age at which sagging expression appears can vary depending on genetics, lifestyle factors, and overall skin health.

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