AE 321 Homework 8 Due in class on November 8, 2013
Formulation of a boundary value problem requires that: a) you identify all boundary conditions on all external surfaces of an object, and for simple surfaces that could be described by a simple outward normal vector, apply Cauchy’s formula to find all stresses on each surface of the object b) identify all body forces c) list all equations that are needed to solve the problem d) briefly describe a strategy to solve the problem, i.e. the sequence in which the equations will be solved and the boundary conditions will be applied. This part is presented in a general and descriptive sense and no equations are actually solved.
Problem 1.
Formulate the following boundary value problem: A cantilever beam of arbitrary but constant cross section (a prismatic beam) is loaded by a shear traction T on its deflecting end, as shown in figure below.
Prismatic beam subjected to a shear traction T
Problem 2. If uniform traction of magnitude T is applied to the beam ends as shown below:
(a) Formulate the 2-D boundary value problem (Identify the BCs on all faces. Assume that the beam has unit thickness in z-direction. Do a stress formulation by making a reasonable assumption for the stresses inside the beam. Then, write all the field equations and describe the procedure to solve the problem. Do not solve the problem in this step.) (b) Solve this problem to find the stresses, strains, and displacements (with some constants)
1
(c) Calculate the displacement field that causes only material deformation assuming that the center of the beam (0,0) undergoes no rigid body rotation about the z-axis or translations in x and y directions. (d) If T is replaced by the statically equivalent force F=2cT use Saint Venant’s principle to provide an estimate of the minimum ratio l /c so that your solution in part (c) is valid at least near the beam centerline.
Problem 3. We wish to calculate the maximum vertical deflection of a beam which is subjected to uniform traction (pressure), w, at its top surface and is supported at its left and right sides at x= ±l with a shear force as shown below:
We will break down and solve this problem following the steps (a) – (e): (a)
Find the resultant shear counter forces that we must apply on the left and the right sides of the beam at x = l to obtain global force and moment equilibrium. Assume that the depth of the beam is 1 (one).
(b)
Formulate the 2-D boundary value problem (assuming that the depth of the beam is 1, no stresses/strains/displacements in the direction of the depth of the beam). Find the boundary conditions at (x, y) = (x, c) using Cauchy’s formula. This will give you 4 equations. Another 3 equations can be constructed at (x, y) = (l , y) by assuming that the resultant normal tractions are zero, the resultant moment due to the normal tractions is zero, and the resultant shear tractions are equal to the statically equivalent tractions calculated in part (a). Use Cauchy’s formula to show that these 3 equations are equivalent to:
2
c
( x l , y)dy 0
( x l , y) ydy 0
( l , y)dy wl
x
c c
x
c c
xy
c
(c)
Assume the following state of stress:
Find the stress, the strain, and the displacement distributions in the beam. To do so use the first 4 boundary conditions to determine some of the unknown constants. Then, verify that the equations c
c
x
( x l , y)dy 0
c
xy
(x l , y )dy wl
c
are satisfied automatically. You could have actually guessed this: If you consider the 4 corner points of the beam as tiny squares they must be under equilibrium. Then use c
x
( x l , y) ydy 0
c
to find the remaining constant. When you solve for the displacements integrate the normal strains first and then apply the solution to the equation of the shear strain. This will give you two equations because x and y are independent variables.) (d)
Assume that u(0,y) = v(l ,0) = 0 (i.e. the vertical beam centerline does not move horizontally and the centers of the left and right beam surfaces do not move vertically) to simplify the displacement equations.
(e)
Find the point where the maximum vertical deflection of the beam occurs and the magnitude of the maximum vertical deflection. Hint: due to symmetry, this point should belong to the vertical centerline.
3
