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Examples of Linear Stiffness - K

In this section some simple examples are presented to show how the stiffness can be determined. Applying a force to a structure and finding the displacement at the point the force is applied, the stiffness can be found from K = f / u.

Uniaxial Bar:

Consider a uniaxial bar which is fixed at the left-end and is subjected to a concentrated force f at the right-end. The material is assumed to behave in accordance to Hooke's law, i.e., = E. The bar is prismatic (A = constant) and homogeneous (E = constant). The displacement at the right-end can be found in any mechanics of materials textbook as u = f L/AE.

The bar can be modeled as a SDOF translational spring if u at the right-end is considered to be the displacement of concern in predicting the response of the bar.

Rearranging the equation above, we obtain: f = (AE/L) u = K u, where A, E, and L are all positive quantities. This is indeed of the form of the force-displacement relationship for a linear spring. Thus we can conclude that for a one-dimensional bar, its stiffness is equilvalent to AE/L. The stiffness of the bar is a function of geometry (A & L) and material property (E). Also since the - law is linear, the force-displacement relationship is linear with stiffness (K = AE/L) being the slope, as shown below.

Torsional Bar:

Consider a shaft which is fixed at the left-end and is subjected to a concentrated torque t at the right-end. The material is assumed to behave in accordance to Hooke's law, i.e., = G . The bar is prismatic (J = constant) and homogeneous (G = constant). The displacement at the right-end can be found in any mechanics of materials textbook as = t L/JG, where t is the torque, L is the shaft length, J is the polar moment of inertia and G is the shear modules.

The shaft can be modeled as a SDOF rotational spring if at the right-end is considered to be the rotation of concern in predicting the response of the shaft.

Rearrange the equation above, we obtain: t = (JG/L) = K_T, where J, G, and L are all positive quantities. This is indeed of the form of the torque-rotation relationship for a torsional spring. Thus we can conclude that for a one-dimensional torsional bar, its stiffness is equilvalent to JG/L. The stiffness of a torsional bar is a function of geometry (J & L) and material property (G). Also since the shear stress-shear strain law ( = G) is linear, the torque-rotation relationship is linear with stiffness (K_T = JG/L) being the slope, as shown below. In this case stiffness K_T is defined as the torque necessary to produce a unit rotation.

Beam:

Consider a simply supported beam which is pinned at the left-end (x and y translational displacements restrained) and is supported by a roller at the right-end (allows translation in x-direction). The beam is subjected to a upward concentrated force f at the center. The material is assumed to behave in accordance to Hooke's law, i.e., = E. The beam is prismatic (I = constant) and homogeneous (E = constant). The displacement at the center can be found in any mechanics of materials textbook as u = f L³/48EI.

The beam can be modeled as a SDOF translational spring if u at the center is considered to be the displacement of concern in predicting the response of the beam.

Rearrange the equation above, we obtain: f = (48 EI/L³) u = K_b u, where E, I and L are all positive quantities. This is indeed of the form of the tranverse force-transverse displacement relationship, where K_b is the bending stiffness. Thus we can conclude that for a beam, its stiffness is equilvalent to 48EI/L³. The stiffness of the beam is a function of geometry (I & L) and material property (E). Also since the - law is linear, the transverse force-transverse displacement relationship is linear with stiffness (K_b = 48EI/L³) being the slope, as shown below.