# Explaining Bending Stress in Cantilever Beams

• Timisoarian
In summary, Bending stress is caused by a bending moment applied to a beam and its magnitude varies along the length of the beam, with a maximum at the fixed end and zero at the free end. This is due to the distribution of the bending moment, which increases as you move towards the fixed end. The bending stress is a combination of tensile and compressive stresses, with tensile stresses considered positive and compressive stresses considered negative. The bending moment can be referred to as hogging or sagging, depending on the direction of the applied force at the free end of the beam.
Timisoarian
Hey everyone,

I was just wondering if someone could explain to me what Bending stress is? Supposed you have a cantilever beam with a point load at the free end, I know that the bending stress will decrease as you move away from the fixed end? but why? and how can you back that up?

Also, as the beam is bending downwards, doesn't that create a hogging moment? tension at the top and compression at the bottom? which mean top is negative bottom is positive? please explain!

Timisoarian said:
Hey everyone,

I was just wondering if someone could explain to me what Bending stress is? Supposed you have a cantilever beam with a point load at the free end, I know that the bending stress will decrease as you move away from the fixed end? but why? and how can you back that up?

Also, as the beam is bending downwards, doesn't that create a hogging moment? tension at the top and compression at the bottom? which mean top is negative bottom is positive? please explain!

You'll need to take a course in mechanics of materials, but basically, the bending moment increases as you approach the fixed end (varies from 0 at free end to maximum PL at fixed end, using moment equation formulae) , and since bending stress is a function of bending moment and beam properties (M(y)/I if you are at all familiar with the formula for bending stress), it too will increase to a maximum at the fixed end topmost or bottom-most fibers. Yes, hogging, tensile bending stress on top fibers, compression bending stress on bottom fibers, no bending stress at neutral axis, etc... what do you mean by negative, that's a loaded word...

Timisoarian said:
Hey everyone,

I was just wondering if someone could explain to me what Bending stress is? Supposed you have a cantilever beam with a point load at the free end, I know that the bending stress will decrease as you move away from the fixed end? but why? and how can you back that up?

The bending stress is caused by a bending moment applied to a beam. Due to the nature of bending and its effects on a beam, the bending stress at at particular location is partly tensile and partly compressive, unlike stresses produced by other types of loadings:

The bending moment in a cantilever beam which has a point load applied at the free end is a maximum at the fixed end and is zero at the free end; thus, the bending stress is a maximum at the fixed end and zero at the free end.

This is a nifty graphic showing a plot of bending moment magnitudes along a cantilever beam:

Also, as the beam is bending downwards, doesn't that create a hogging moment? tension at the top and compression at the bottom? which mean top is negative bottom is positive? please explain!

In most stress conventions, tensile stresses are considered positive, while compressive stresses have negative magnitudes.

If the force applied at the tip of the cantilever is pointing down, then the bending moment can be called "hogging" [which term is mostly used in nautical circles].

If the force applied at the tip of the cantilever is pointing up, then the bending moment can be called "sagging".

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## 1. What is bending stress in cantilever beams?

Bending stress in cantilever beams is a type of stress that occurs when a load is applied perpendicular to the length of the beam. This causes the beam to bend, resulting in tension on one side and compression on the other. Bending stress is an important factor to consider in the design and analysis of cantilever beams.

## 2. What causes bending stress in cantilever beams?

Bending stress in cantilever beams is caused by external loads acting on the beam, such as forces or weights. The magnitude and location of these loads can greatly affect the amount of bending stress on the beam. Additionally, the length and cross-sectional shape of the beam also play a role in determining the bending stress.

## 3. How is bending stress calculated in cantilever beams?

The bending stress in cantilever beams can be calculated using the formula σ = (M * c)/I, where σ is the bending stress, M is the bending moment, c is the distance from the neutral axis to the outermost fiber, and I is the moment of inertia of the beam's cross-sectional area. This formula takes into account the load and geometry of the beam to determine the maximum stress at a given point.

## 4. What are the factors that affect bending stress in cantilever beams?

The factors that affect bending stress in cantilever beams include the magnitude and location of external loads, the length and cross-sectional shape of the beam, and the material properties of the beam. Additionally, the support conditions of the beam, such as whether it is fixed or pinned, can also impact the amount of bending stress.

## 5. How can bending stress in cantilever beams be reduced?

Bending stress in cantilever beams can be reduced by changing the beam's geometry, such as increasing the cross-sectional area or decreasing its length. Additionally, using materials with higher strength and stiffness can help to minimize bending stress. Properly supporting the beam and distributing the load evenly can also help to reduce bending stress.

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