Beam Analysis and Beam Deflections
In Strength of Materials , you were introduced to several methods of determining beam deflections, which
are required in order to analyze statically indeterminate beam problems. Two of the more useful methods,
Singularit
Statically Indeterminate Examples Using Castiglianos Theorem
Example 1: For the propped-cantilever beam, shown,
w 2 kip/ft , L 8 ft , and EI 135 106 lb-in 2 . Considering only
the strain energy due to bending, determine the reaction at B using
Castigilano
Preliminary Analysis of Spring Stiffness for a Prosthetic Foot
In some cases an amputee requires the use of a prosthetic foot to assure mobility. The overall analysis of
such a device is quite complex and requires knowledge of an individual patients inten
Shear Centers
The beam shown is subjected to a vertical end load. The
load is applied so that it passes through the centroid ( z ) of
the beam. The beam will be subjected to a normal stress
due do bending. In addition, because of the location of the
load,
Eccentric Loads Secant Formula
The Euler formula was derived based on
the assumption that the load P was
always applied through the centroid of
the columns cross-sectional area and
that the column is perfectly straight. This
is an unrealistic situation si
Safety Factors Another Approach
Puggley (1966) uses a somewhat different approach to defining a design safety factor ( nd or ns ). Defining the
allowable normal stress as all (typically the yield stress, S yp ) and the design normal stress as d , Pugsley
Non Standard Sections Examples
Example 1
The section shown is subjected to a torque T. It is made of an alloy for
which the yield point in shear is 86 ksi. If a design factor of safety of
nd 2 is required, determine the largest torque that can be applied.
Impact Examples AU 2012
Example 1
A weight ( w 300 lb ) is dropped from a height h onto
a steel ( E 30 106 psi , S yp 36 ksi ). Determine the
height h from which the weight can be dropped without
causing yielding.
Solution
From Table 4.1, Case 2 M max Pab
Curved Beams
We will use the same notation as used in your text. Following is a summary, also see
Figure 4.29.
r
radial coordinate of an arbitrary beam element
y
coordinate of an arbitrary element measured from the neutral axis
rc r radial location of the
Castiglianos Theorem
Castiglianos theorem states that the displacement of point i in a structure (generally a beam) can be
determined by taking the partial derivative of the strain energy of the system with respect to the load
applied at point i. In other
Buckling Examples AU 2012
Example 1
Consider a 3-m long column with fixed ends and a square cross section with unknown
dimensions. We wish to apply a 100 kN force to the column and have a safety factor
of 3 ( n 3 P / Pall ), where the allowable load is Pa
St r ess a n d St r a i n T r a nsf o r m a t io n E q u a t io ns a n d St r a i n G a ges : T r a nsf o r m a t io n E q u a t io ns
A general state of stress is given in the figure to the right. For a state of plane
0 . The resulting state of stress is