Strength and Stiffness in Mechanics of Materials: The Best Explanation in History

Many people are always confused about the concepts of strength and stiffness in mechanics. Let’s talk about our understanding today.


In order to ensure the normal operation of the mechanical system or the whole structure, each part or component must be able to work normally.

The task of safety design of engineering components is to ensure that the components have sufficient strength, stiffness and stability.

Stability is well understood, the ability to maintain or restore the original equilibrium form under force.

For example, the thin rod under pressure suddenly bends, the load-bearing of thin-walled members folds, or the instability of building columns leads to collapse, which is well understood.

Today, I will mainly talk about my understanding of stiffness and strength.



What is strength

Definition: the ability of components or parts to resist damage (fracture) or significant deformation under the action of external force.

Extract keywords, damage, fracture, significant deformation.

For example, Tom regarded the iPad as a scale. When he stood up, the iPad screen cracked, which is not strong enough.

For example, when at summer, many big branches are broken by the wind, which is not strong enough.

Strength is a parameter that reflects the failure of materials such as fracture.

Strength generally includes tensile strength and compressive strength, which is the amount of material failure when the stress reaches.

The unit of strength is generally MPa.

Failure type of strength

Brittle fracture: a sudden fracture that occurs without obvious plastic deformation.

For example, the fracture of cast iron specimen along the cross section during tension and the fracture of cast iron specimen with circular section along the oblique section during torsion.

Plastic yield: the material produces significant plastic deformation and makes the component lose its working capacity.

For example, the low-carbon steel sample will have significant plastic deformation during tension or torsion.

Strength theory

1. Maximum tensile stress theory:

As long as the maximum tensile stress σ1 at one point in the member reaches the ultimate stress σb under the unidirectional stress state, the material will undergo brittle fracture. Therefore, the conditions for brittle fracture failure of components with dangerous points in complex stress state are as follows: σ 1= σ b。

Therefore, the strength conditions established according to the first strength theory are: σ 1≤[ σ] 。

2. Maximum tensile strain theory:

As long as the maximum tensile strain ε1 reaches the limit valueε u under unidirectional stress state, the material will undergo brittle fracture failure. ε 1= σ u;

From the generalized Hooke’s Law: ε 1=[ σ 1-u( σ 2+ σ 3) ] / E, so σ 1-u( σ 2+ σ 3)= σ b。

The strength conditions established according to the second strength theory are: σ 1-u( σ 2+ σ 3)≤[ σ]。

3. Maximum shear stress theory:

As long as the maximum shear stressτMax reaches the ultimate shear stress τ0 in the unidirectional stress state, the material will yield failure. τ max= τ 0

According to the stress formula on the inclined section of axial tension τ 0= σ s/2( σ S – normal stress on the cross section) is obtained from the formula: τ max=( σ 1- σ 3)/2。 So the damage condition is rewritten as σ 1- σ 3= σ s。

The strength condition according to the third strength theory is: σ 1- σ 3≤[ σ]。

4. Shape change specific energy theory:

As long as the shape change ratio at a point in the member can reach the limit value under the unidirectional stress state, the material will yield failure.

Therefore, the strength condition according to the fourth strength theory is:

sqrt( σ 1^2+ σ 2^2+ σ 3^2- σ one σ 2- σ two σ 3- σ three σ 1)<[ σ]。

2. Stiffness

What is stiffness

Definition: refers to the ability of members or parts to resist elastic deformation or displacement under the action of external force, that is, the elastic deformation or displacement shall not exceed the allowable range of the project.

Stiffness is a parameter reflecting the relationship between structural deformation and force, that is, the amount of deformation generated by how much force the structure is subjected to.

In short, it is a spring, and the stiffness of the spring is the tensile force divided by the elongation. The unit of stiffness is generally N / m.

Stiffness type:

When the applied load is a constant load, it is called static stiffness;

When it is an alternating load, it is called dynamic stiffness.

Static stiffness mainly includes structural stiffness and contact stiffness.

Structural stiffness refers to the stiffness of the member itself, mainly including bending stiffness and torsional stiffness.

1. Bending stiffness: calculated according to the following formula:


Where P — static load (n);

δ—— Elastic deformation in load direction( μm)。

2. The torsional stiffness is calculated according to the following formula:


Where M — applied torque (n · m);

L — distance from the torque action position to the fixed end (m);

θ—— Torsion angle (°)

3. Relationship between strength and stiffness

strength vs stiffness

Through the above theoretical understanding of strength and stiffness, relative to stiffness, the definition of strength is aimed at the failure under the action of external force, and the failure types are classified as plastic yield and brittle fracture, which is associated with the stress-strain curve during tension.

As shown in the fig.

Relationship between strength and stiffness

The curve in the figure can be divided into four stages:

1. Elastic deformation stage;

2. Yield stage;

3. Strengthening stage;

4. Local necking stage.

The definition of stiffness is to resist elastic deformation, which is carried out in the first stage. Hooke’s law is satisfied under elastic action.

Observe the calculation formula of bending stiffness and torsional stiffness under static load, which is similar to Hooke’s law.

It can be inferred that the measurement of stiffness is only carried out in the elastic deformation stage.

After entering the next stage, for the plastic strain in the tensile process, the residual strain of fire will not disappear.

Under the stress-strain curve, the stress is almost unchanged, but the strain increases significantly.

At this time, the stress is the yield limit. For the material, it enters the failure stage of plastic yield. After entering the strengthening stage, the strain increases with the increase of stress, and finally reaches the strength limit.

It can be seen that the measurement of strength is after the elastic deformation of the material and before the strength limit.

Wrap it up

In conclusion, it can be concluded that both stiffness and strength are measured in the failure stage of parts, and stiffness can be measured by stress, and strength can be measured by deformation.

In the strain process, stiffness is in the previous stage and strength is in the later stage.

Therefore, in the measurement of failure conditions of parts, as long as the stiffness requirements are met, enough stress can be resisted in the elastic deformation stage.

On this premise, the strength meets the requirements of parts.

According to this relationship, there will be various designs in actual production, such as the shaft in mechanical equipment.

Usually, the size of the shaft is determined according to the strength conditions, and then the stiffness is checked according to the stiffness conditions.

Therefore, the stiffness requirements of precision machinery for the shaft are set very high, and the design of its section size is often controlled by the stiffness conditions.

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