Stiffness Formulas

Spring Stiffness and Hooke's Law

Everything is a spring and every spring has linear behavior up to a point.

Hooke's Law tells us that the force varies as the stretch: FORCE = STIFFNESS x DEFORMATION or F = kx. We can also focus on stiffness: STIFFNESS = FORCE / DEFORMATION or k = F/x. Anything that requires more force to deform is considered stiff. Anything that requires less force to deform is considered flexible. Flexibility is simply the inverse of stiffness.

Hooke's Law applies to an individual spring and anything else that resists deformation under load, including materials, structural members, and complex assemblages of structural members.

Member Stiffness

At the member level, we usually first encounter Hooke's Law in terms of an axial spring. If a spring stretches 1 inch when we apply 1 pound, Hooke's Law tells us that if we apply 2 pounds then the spring will stretch 2 inches. The spring's stiffness, or the spring constant, is k = 1 lb/in.

This is the linear response of structures that we can expect unless other behavior intercedes like yielding of the materials that make up the member, non-linear material behavior, large deformations, or instability that can lead to collapse.

Natural Frequency

Other parts of this website are dedicated to the relationship of stiffness and the mass of a structure that it supports. The arrangement of stiffness and mass in a structure results in a natural frequency of the structure, which is a fundamental property of the structure, like a fingerprint, that can be used to validate models of the structure, either analytical (a mathematical formula) or numerical (a computer-based model).

Stiffness is an important aspect of structures that shows up in both static loading situations (like Hooke's Law) and dynamic situations (when we experience vibrations and the natural frequency of a structure).

Static Response to Load

A 10-lb weight was suspended from a spring of unknown stiffness.

The initial position can be seen as roughly 1-1/8".

An additional 5-lb weight was added, displacing the weights to 2-1/2", which stretched the spring by 1-3/8".

The spring constant is then k = F/x = (5 lb)/(1.375 in) = 3.64 lb/in.

Dynamic Response: Free Vibration

We can set the 15 lbs oscillating in free vibration and measure the natural frequency. The total suspended mass can be calculated as m = W/g = (15 lb)/(386.4 in/s2) = 0.03882 lb-s2/in. The natural frequency of the suspended 15 lb depends on the stiffness (3.86 lb/in) and mass (0.03882 lb-s2/in) and can be calculated as 1.54 Hz.

An iPhone 13 Pro running the PhyPhox app with the front-facing LiDAR sensor was placed below the weight.

Using the Closest setting ensures that the elements nearest the sensor are used for measurement.

Six cycles is measured at 3.940 seconds, so the natural period is 0.6566 seconds, and the natural frequency is the inverse of the period. So, the natural frequency has been measured as 1.52 Hz, which is roughly 1% different than the calculation.

If we work backwards through this experiment, it is clear that we can use a measured natural frequency of structural system to determine its stiffness, so long as its weight is known. In the case of the spring system here, the dynamic measurement of 1.52 Hz results in a stiffness of 3.54 lb/in. This is less than the 3.64 lb/in that was measured statically and is a common result for systems measured dynamically, primarily due to motion changing directions.

These ideas are exploited throughout this website to learn more about structures with increasing levels of complexity. A simple single-element model can be created in MASTAN to demonstrate both the static and dynamic behavior of this spring-mass system. With the stiffness of 3.54 lb/in employed in the model, a natural frequency of 1.52 Hz is recovered, confirming that the computer model is producing the same result as our physical model.

Next, let's examine what contributes to the stiffness of structural members.

Stiffness and Flexibility of Structural Members

We can consider structural members as simple springs. But rather than have a coil that gives the spring its flexibility, it is the geometric and material properties of the member that dictate its stiffness.

Specifically,

if the cross-sectional area is large (like the column of a building),
the material itself is stiff (like concrete),
and the member is short,

the member will be stiff and will not deform substantially under load.

Conversely,

if the cross-section area is small (like a thin wire),
the material is less stiff (like a rubber band),
and the member is long,

the member will be less stiff, or more flexible.

By considering properties in this way it makes sense that

F = kx becomes F = (AE/L)x

where

A is the cross-sectional area,

E is the modulus of elasticity (the axial stiffness of the material), and

L is the length of the member.

Four Types of Internal Force

In structural mechanics, we develop these relationships for each of the four forces a structural member can carry: axial load, shear force, torsion, and bending moment. These are depicted here with a summary of the relevant formulas that you would learn in a first course in solid mechanics.

We can see that if a member carries axial force, its stiffness is AE/L, and its deformation is a stretch or a change in length.

If a member carries shear force, its stiffness is AG/L, and its deformation is a displacement in the direction of the shear force.

If a member carries torque, its stiffness is JG/L, and its deformation is a twist along the length of the member.

And, if a member carries bending, its stiffness is EI/L, and its deformation results in curvature and displacement transverse to the length. The specific support conditions and loading on a beam lead to many force-deformation scenarios that are often tabled for ease of use.

A, J, and I are cross-sectional properties that define the section's stiffness relative to axial (A), shear (A), torsion (J), or bending (I) forces. These properties can be calculated for simple cross sectional shapes and are tabled for many standard shapes.

E and G are material properties that define the material's stiffness. E is the modulus of elasticity that relates axial stress and axial strain up to yield. G is the shear modulus of elasticity or modulus of rigidity that relates shear stress and shear strain.

Material Stiffness

Material stiffness is a relationship of stress and strain. Stress is the intensity of a force at a point in a member and strain is the intensity of deformation at a point in a member. Because the intensity of force and deformation are considered at a point, they do not depend on the geometry of the member, but only on its material. Thus, Hooke's Law applies at the material level and the material stiffness is a material property:

Modulus of Elasticity, E = Axial Stress / Axial Strain

Modulus of Rigidity, G = Shear Stress / Shear Strain

System (or STructure) Stiffness

Structures are composed of structural members that can carry axial force, shear force, torsion, or bending along their lengths. If all of these forces are considered and members are arranged in three dimensions, the stiffness of the structure can be described by combining the member stiffnesses according to the connection of the members and their support conditions. This is done using stiffness matrices and is explored further on the Matrix Structural Analysis page of this website.

The graphic below outlines the three levels of Hooke's Law.

1) At the material level, Hooke's Law relates stress and strain.

2) At the member level, Hooke's Law relates force and deformation.

3) At the system or structure level, Hooke's Law relates loads applied to a structure and the resulting displacements. At this level, the analysis of the structure requires the use of static equilibrium to relate loads to reactions at supports and to relate loads to the internal forces in the various members composing the structure. The member force-deformation relationships - or member stiffnesses described above - are also required. Finally, the relationship of member deformations to the displacements of the structure, which is called compatibility, must be described.

References

Seeing Structures by Susan Reynolds

MecMovies by Timothy Philpot

Multimedia Mechanics of Materials by Kurt Gramoll

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