Understanding of strength and stiffness

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Many people often get confused about the concepts of strength and stiffness in mechanics; today I’ll share my own understanding. The book states that in order to ensure the proper functioning of a mechanical system or an entire structure, each of its components or parts must be able to work properly. The task of safe design for engineering components is to ensure that they possess sufficient strength, stiffness, and stability. Stability is easy to understand: it is the ability to maintain or return to the original balanced state under external forces. For example, a stressed thin rod may suddenly bend, thin-walled components may wrinkle under load, or building columns may become unstable and cause the structure to collapse. Today, we will mainly discuss the understanding of stiffness and strength. 01 Strength Definition: The ability of a component or part to resist destruction (fracture) or significant deformation under the action of external forces. For example, if someone uses an iPad as a weight scale and steps on it, the screen of the iPad cracks – that’s because its strength isn’t sufficient. For example, in Wuhan, many tree branches break in the summer due to strong winds, which is also a result of insufficient strength. Strength is a parameter that indicates when a material undergoes failure such as fracturing. Common types of strength include tensile strength and compressive strength, which represent the level of stress at which the material fails. The unit of strength is generally megapascals. 1.1 Type of failure: Fracture – a sudden break that occurs without significant plastic deformation. Such as the fracture of cast iron specimens along their cross-section during tensile testing, and the fracture of circular-section cast iron specimens along an inclined section during torsional testing. Plastic yield occurs when a material undergoes significant plastic deformation, resulting in the loss of its functional capacity; for example, low-carbon steel specimens exhibit significant plastic deformation under tension or torsion. 1.2 Strength theory – Maximum tensile stress theory: As long as the maximum tensile stress σ1 at a certain point within the component reaches the ultimate stress σb in a uniaxial stress state, the material will undergo brittle fracture. Thus, the condition for brittle fracture failure of a member under a complex stress state at the critical point is: σ1=σb. Therefore, the strength condition established based on the first strength theory is: σ1≤ . According to the theory of maximum tensile strain, as long as the maximum tensile strain ε1 reaches the limit value εu in a uniaxial stress state, the material will undergo brittle fracture failure, with ε1=εu. From the generalized Hooke’s law, we have: ε1 = E/σ1, so σ1 – u(σ2 + σ3) = σb. The strength condition established based on the second strength theory is: σ1 – μ(σ2 + σ3) ≤. According to the maximum shear stress theory, as long as the maximum shear stress τmax reaches the ultimate shear stress τ0 in a uniaxial stress state, the material will yield and fail. τmax=τ0. According to the stress formula for a shear section under axial tension, τ0 = σs/2 (where σs is the normal stress on the cross-section). From this formula, it follows that 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≤. The theory of shape change specific energy states that as long as the shape change specific energy at a certain point within a component reaches the limit value under uniaxial stress conditions, the material will yield and fail. Therefore, the strength condition according to the fourth theory of strength is as follows: Stiffness Definition: It refers to the ability of a component or part to resist elastic deformation or displacement under external forces; in other words, the elastic deformation should not exceed the limits permitted by engineering standards. Stiffness is a parameter that reflects the relationship between structural deformation and the magnitude of force, that is, it indicates how much deformation a structure undergoes under a given force. Simply put, for a spring, its stiffness is equal to the pulling force divided by the amount of stretch it experiences. The unit of stiffness is generally N/m. 2.1 Stiffness type When the applied load is a constant load, it is referred to as static stiffness ； For alternating loads, it is referred to as dynamic stiffness. Static stiffness mainly includes structural stiffness and contact stiffness. Structural stiffness refers to the stiffness of the components themselves, which primarily consists of bending stiffness and torsional stiffness. The bending stiffness is calculated using the following formula: where P is the static load (N), and δ is the elastic deformation in the direction of the load (μm). The torsional stiffness is calculated using the following formula: where M is the applied torque (N·m), L is the distance from the point where the torque acts to the fixed end (m), and θ is the torsional angle (°). 03 Relationship between the two: Through the theoretical understanding of strength and stiffness mentioned above, compared to stiffness, the definition of strength refers to failure under the action of external forces ； The failure types are classified as plastic yield and brittle fracture, which leads to the consideration of the stress-strain curve during tension testing. As shown in the figure. Image: The curve in the image can be divided into four stages: I, elastic deformation stage ； II. Yield stage ； III. Strengthening Phase ； IV. Local necking stage. The definition of stiffness is the ability to resist elastic deformation, and this occurs in the first stage; under elastic forces, Hooke’s law holds true. By examining the formulas for calculating bending stiffness and torsional stiffness under static loads, which are similar to Hooke’s law, it can be inferred that the measurement of stiffness takes place only during the elastic deformation stage. Upon entering the next stage, the plastic or residual strain generated during stretching does not disappear; on the stress-strain curve, the stress remains almost constant while the strain increases significantly. At this point, the stress corresponds to the yield limit, and the material enters a phase of plastic yielding failure. After entering the strengthening stage, strain increases as stress increases, eventually reaching the strength limit. It can be seen that the measurement of strength is carried out after elastic deformation of the material but before the strength limit. In summary, it can be concluded that both stiffness and strength are measured values related to the failure stage of a component; stiffness can be determined based on stress, while strength can be determined based on deformation. During the strain process, stiffness is relevant in the previous stage while strength is important in the subsequent stage; therefore, when measuring the conditions for part failure, as long as the stiffness requirements are met, sufficient stress can be resisted during the elastic deformation phase, and under such circumstances the strength requirements for the part are also satisfied. Based on such a relationship, various designs are developed in actual production; for example, in mechanical equipment, the dimensions of shafts are first determined based on strength requirements, and then their stiffness is checked according to stiffness criteria. Therefore, precision machinery imposes very high requirements on the stiffness of shafts, and the design of their cross-sectional dimensions is often determined by stiffness constraints.