Many people are always confused about the concepts of strength and stiffness in mechanics, so today I’ll talk about my understanding.

Table of Contents

**Overview**

**Overview**

The book says that in order to ensure the proper functioning of a mechanical system or the entire structure, each component or element must be able to function properly.

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

Stability is well understood as the ability to maintain or restore the original equilibrium form under the influence of a force.

For example, the thin rod under pressure suddenly bends, the load-bearing thin-walled member folds or the column of the building collapses, which is well understood.

Today, we will focus on the meaning of stiffness and strength.

**I. Strength**

**I. Strength**

Definition: the ability of a component or part to resist damage (fracture) or significant deformation under the influence of an external force.

Keywords: damage fracture, significant deformation.

For example, a fat man uses the iPad as a weight scale, and when he stood upon it, the iPad screen was cracked, which was insufficient strength.

For example, many large tree branches are blown off by the wind every summer, which is also insufficient strength.

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

Strength generally includes tensile strength, compressive strength, etc., which is the amount of material damage when the stress reaches, and the strength unit is generally MPa.

**Damage type**

**Damage type**

** Brittle fracture: **A sudden fracture that occurs without significant plastic deformation.

For example, the fracture of cast iron specimens along the cross section when stretched and the fracture of round section cast iron specimens along the oblique section when twisted.

** Plastic yield:** Significant plastic deformation produced by plastic yield makes the component lose the working ability, e.g., a mild steel specimen undergoes significant plastic deformation in either tension or torsion.

**Strength theory**

**Strength theory**

**Theory of maximum tensile stress:**

As soon as the maximum tensile stress σ1 at a point inside the member reaches the ultimate stress σb in a unidirectional stress state, the material will undergo brittle fracture.

Thus, the condition for the brittle fracture failure of a component at the danger point in a complex stress state is σ1 = σb.

Therefore, the strength condition established according to the first strength theory is: σ1≤[σ].

**Maximum tensile strain theory：**

As soon as the maximum tensile strain ε1 reaches the limit value of εu in the unidirectional stress state, the material will undergo brittle fracture failure. ε1 = σu.

From the generalized Hooke’s law: ε1 = [σ1-u(σ2+σ3)]/E, so σ1-u(σ2+σ3) = σb.

The strength condition established by the second strength theory is: σ1-u(σ2+σ3) ≤ [σ].

**Theory of maximum shear stress.**

As soon as the maximum shear stress τmax reaches the ultimate shear stress τ0 in the unidirectional stress state, the material will undergo yield failure. τmax=τ0.

According to the formula of stress on the axial tensile inclined section, we can know that τ0=σs/2(σs – positive stress on the cross-section) from the formula: τmax=(σ1-σ3)/2.

So the failure condition is rewritten as σ1-σ3=σs.

The strength condition according to the third strength theory is: σ1-σ3 ≤ [σ].

**Shape change specific energy theory**

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

Therefore, the strength conditions according to the fourth strength theory are.

sqrt(σ1^2 + σ2^2 + σ3^2 – σ1σ2 – σ2σ3 – σ3σ1) < [σ].

**II. stiffness**

**II. stiffness**

Definition: the ability of a component or part to resist elastic deformation or displacement under the action of an external force, i.e. elastic deformation or the only range that should not exceed the engineering allowance.

Stiffness is a parameter reflecting the relationship between the deformation of a structure and the magnitude of the force, i.e. the amount of deformation produced by how much force the structure is subjected to.

Simply, it is a spring, tension divided by elongation is the stiffness of the spring.

The stiffness unit is generally N/m.

**Types of Stiffness**

**Types of Stiffness**

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.

The structural stiffness is the stiffness of the member itself, mainly including bending stiffness and torsional stiffness.

**B**calculated by the following formula:**ending stiffness:**

where P – static load (N);

δ – elastic deformation in the load direction (μm).

calculated by the following formula:**Torsional stiffness:**

where M – the torque acting (N-m);

L – distance from the torque acting point to the fixed end (m);

θ – torsional angle (°).

**III. Linkages between the two**

**III. Linkages between the two**

Through the above theoretical understanding of strength and stiffness, the definition of strength, as opposed to stiffness, addresses the destruction by external forces, while the types of destruction are classified as plastic yielding and brittle fracture, which leads to the stress-strain curve in tension.

This is shown in the following figure.

The curve can be divided into four stages.

- Elastic deformation stage.
- The yield stage.
- Strengthening stage.
- IV, local necking stage.

Whereas the definition of stiffness lies in the resistance to elastic deformation and is carried out under the first stage, satisfying Hooke’s law in elastic action.

Observing the formulas for calculating bending and torsional stiffness under static load, similar to Hooke’s law, it can be inferred that the stiffness is measured only in the elastic deformation phase.

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

Under the stress-strain curve, the stress is almost unchanged and the strain increases significantly, and this stress is the yield limit.

The material enters the plastic yield failure phase, and after the strengthening phase, the strain increases with increasing stress and finally reaches the strength limit.

Thus, the strength is measured after the elastic deformation of the material and before the strength limit.

In summary, it can be concluded that both stiffness and strength are measured at the failure stage of the part.

Whereas stiffness can be measured by stress and strength can be measured by deformation, with stiffness in the first stage while strength in the second stage of the strain process.

Therefore, in the measurement of the failure condition of the part, as long as the rigidity requirement is met, it can resist enough stress in the elastic deformation stage, and the strength also meets the requirements of the part under this premise.

It is in accordance with such a relationship that there are designs of various kinds in actual production, such as shafts in mechanical equipment.

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

The stiffness requirements for shafts in precision machinery are thus set very high, and the design of their cross-sectional dimensions is often controlled by the stiffness conditions.