Equilibrium is achieved if all the forces acting on an object are balanced. The second equation is satisfied automatically, and we observe that the stresses in an end-loaded cantilever beam as found from elementary beam analysis do satisfy the equations of equilibrium. Some everyday examples of equilibrium include: a car at rest at a stop sign, a car moving at a constant speed, two people balancing on a see-saw, two objects at equal temperature, two objects with the same charge density and the population of a species staying the same. Substituting these two results into the first equation of equilibrium gives If the beam cross section is rectangular with width b and height 2 c, we obtain. In the following V is the shear force at position x, and Q is the first moment of the cross sectional area above the point y where the shear stress is to be found. We can calculate the shear stress using element beam theory. ![]() Again, the equilibrium equations are satisfied when s x is substituted.Įxample 5-2-3 For the end-loaded cantilever beam shown, the moment varies linearly along the length of the beam giving the following for the bending stress. The axial stress is given by s x = P/ A all other stresses are zero, and you see by substitution into the above equations that equilibrium is satisfied.Įxample 5-2-2 For pure bending of a beam by a moment M that does not vary along the length of the beam, the bending stress is given by s x = -My/ I, and all other stresses are zero. The correct stress distribution for a problem in solid mechanics must satisfy these equations of equilibrium.Įxample 5-2-1 Consider an axially loaded, uniform rod. Writing the equation of equilibrium for the stresses in the y direction produces a similar result.Ĭonsidering a three-dimensional case in which stresses act in the x, y, and z directions gives the following set of equations of equilibrium. Teacher dashboard to automate grading and track. 250+ Web-based simulations that can be played on laptops and tablets without installing any software. ![]() Simulate experiments, train lab techniques, and teach theory through visual experiences that enhance long-term learning outcomes. If there is an element force, X b, in the x direction that is proportional to the volume (a ‘body force’), such the weight or an electromagnetic force, it must be added to the above. Engage students in science through interactive learning scenarios. (For simplicity only the x normal and xy shear stresses are shown.) Since the stresses change as one moves from point to point within the body, a variation in each stress term is considered using a first order Taylor series expansion. Z, from a linear elastic body as shown in the figure, we can write the equations of equilibrium for the stress components acting on the faces of the element.
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