What does a plane stress element indicate

Background A previous article explained that even with powerful modern computers, there is often a motivation to use simplifying techniques in Structural FE Analysis. This follow-on article describes how two closely related methods can be used to take 2D slices through a complex structure at regions of interest. The resulting FEA models can give valuable insight into local stresses more rapidly and efficiently than a full 3D model. They won’t tell the whole story – but are valuable tools for the CAE engineer. The two FEA methods are Plane Stress and Plain Strain. Both use 2D planar elements that look very much like thin shell elements and are meshed using planar surface geometry. We will review the background to each method and illustrate with some examples of how useful they are in real world simulation. Plane Stress Analysis Figure 1 shows the important facts about Plane Stress Analysis. The structural region is assumed to lie in the 2D xy plane, with the third structural dimension relatively small. In the figure this is the thickness in the z direction. Stresses exist in the 2D plane as sigma x, sigma y (direct stresses) and sigma xy (in-plane shear stress). Each of these stresses is constant through the thickness as shown in the inset. In addition there can be no stress in the z direction. This stress-strain material relationship is defined in 2D Plane Stress elements used in this type of analysis. Figure 1. Plane Stress. Stress state assumptions The lack of z STRESS i...

Let $$\sigma=\left(\beginz=k$ where $k$ is a constant. Since this stress tensor does not depend on any $x,y,z$ am I right in assuming I can set this constant to be zero for simplicity? Then do I multiply the tensor and normal vector together to get the vector on the plane? Then just trigonometry it to get the angle? The stress vector $\boldsymbol\right).$$

\( \newcommand holds. Note The above example an serve as a practical application of the Saint-Venant’s principle (1856). This principle named after the French elasticity theorist, Jean Claude Barre’ de Saint-Venant can be stated as: “the difference between the effects of two different but statically equivalent loads become very small at sufficiently large distances from load.” Think what are the “two” equivalent loads that are applied to the bar ends? We usually think of a cross-section being cut perpendicular to the axis of the bar. Consider now two cuts at the angles \(\theta\) and \(\left(\frac will be derived in a different way. Symmetry of the stress tensor It should also be noted from Equation \ref\): Components of the stress tensor on three facets of the infinitesimal surface element. Sign convention The Cauchy formula can also be consistently used to determine the sign of the components of the stress tensor. The point is that the sign of the components of the vectors is known from the chosen coordinate system. For illustration, let us orient the volume element along the \(x_1\) axis. With positive direction to the right. Figure \(\PageIndex\). The sign convention is opening the way for deriving the equations of equilibrium for the 3-D continuum. This topic is the subject of the next section. Equilibrium The equilibrium equation for an infinitesimal volume element are derived first using two methods. Referring to Figure (\(\PageIndex + B_2 = 0\] Invoking the index n...

Mechanics eBook: Plane Stress Ch 7. Stress Analysis Multimedia Engineering Mechanics Stress Stresses for Stress Vessels Mechanics Plane Stress New eBook website Chapter 1. 2. 3. 4. 5. 6. 7. 8. 9. Appendix eBooks Author(s): Kurt Gramoll ©Kurt Gramoll MECHANICS - THEORY Combined Stress (or Loads) Stresses at a point (Stress Element) for a Cantilever Beam In the previous sections, both the Stress Element Stresses at a point (Stress Element) for Pressurized Pipe The beam above is just one possible configuration for multiple stresses acting at a point. Another possibility is a pipe that is pulled, pressurized and twisted at the same time. These three loads on the pipe will cause tension normal stress in both directions (axial and circumferential) and cause a twisting or shear stress. All three loads and their associate stresses can be combined together to give a total stress state at any point. The stress element for this example is shown at the left. This section will examine a stress element to better understand stresses at a point and how they can be analyzed. Sign Convention for Stress Element Sign Convention for Stress Element Positive Directions The sign convention for stresses at a point is similar to other stresses. Normal tension stress in both the x and y direction are assumed positive. The shear stress is assumed positive as shown in the diagram at the left. Shear stress act on four sides of the stress element, causing a pinching or shear action. All shear stresses o...