# The virtual principles for simple beams

• B. E. Gatewood
Part of the Mechanics of Structural Systems book series (MSSY, volume 6-7)

## Abstract

The beam in one plane is regarded as a simple beam. Take the particular case in Figure 2.1 with forces in the yz plane only and bending moments about the χ;-axis only. See Figure 4.1 for the notation and loading for the simple beam. From Equations (1.93) and (2.32) the stress, strain, and displacement relations for the beam in Figure 4.1 are

$$\begin{gathered} \sigma _{yy} = Ee_{yy} = - Ez\frac{{{\rm d}^2 u_{zb} }} {{{\rm d}y^2 }} = - \frac{{M_x z}} {{I_{xx} }},\,\,\,\,u_z = u_{zb} + u_{zs} \hfill \\ \theta _x = \frac{{{\rm d}u_{zb} }} {{{\rm d}y}},\,\,\,\sigma _{yz} = Ge_{yz} = G\frac{{{\rm d}u_{zs} }} {{{\rm d}y}} = \frac{{V_z }} {{A_s }}, \hfill \\ \end{gathered}$$
(4.1)
$$\begin{gathered} u_z = u_{zL} \,at\,y = L,\,\,\,\,u_z = u_{z0} \,at\,y = 0, \hfill \\ \theta _x = \theta _{xL} \,at\,y = L,\,\,\,\,\theta _x = \theta _{x0} \,at\,y = 0. \hfill \\ \end{gathered}$$
(4.2)

## Keywords

Cantilever Beam Beam Element Unit Load Virtual Displacement Moment Curve
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• B. E. Gatewood
• 1
1. 1.Dept. of Aeronautical and Astronautical EngineeringThe Ohio State UniversityColumbusUSA