Calculating Bending Moment Using Simpson's 1/3 Rule
What is the formula for calculating the bending moment on a beam using Simpson's 1/3 rule?
The formula for bending moment M in a beam subjected to a shear force V(x) is given by: M = (h/3) * [V(0) + 2 * ∑V(even) + 4 * ∑V(odd) + V(last)].
What is the shear force function V(x) given in the problem?
The shear force function V(x) is given as: V(x) = 5 + 0.25x².
How many segments are used in the 1/3 Simpson's integration rule to evaluate the bending moment?
The 1/3 Simpson's integration rule uses 8 segments to evaluate the bending moment on the beam.
Formula for Calculating Bending Moment Using Simpson's 1/3 Rule
The formula for calculating the bending moment M in a beam subjected to a shear force V(x) using Simpson's 1/3 integration rule is: M = (h/3) * [V(0) + 2 * ∑V(even) + 4 * ∑V(odd) + V(last)].
Shear Force Function V(x)
The shear force function V(x) given in the problem is: V(x) = 5 + 0.25x².
Segments in 1/3 Simpson's Integration Rule
The 1/3 Simpson's integration rule uses 8 segments to evaluate the bending moment on the beam.
The question requires the use of Simpson's 1/3 rule for numerical integration to find the bending moment M for a beam subjected to a shear force V(x). Following the rule's formula with the given shear force function and using the specified eight segments yields the bending moment at the beam's end.
The subject of this question involves the concept of shear force and bending moment in a beam, which is typically studied within the realm of engineering, specifically in the structural and mechanical disciplines. To find the bending moment M(x) at any point x along the beam, we need to integrate the shear force function V(x) = 5 + 0.25x² from 0 to 4.5 meters.
As instructed to use the Simpson's 1/3 rule for numerical integration with 8 segments, we will be dividing the beam's length into 8 equal parts, each of length h = (4.5 - 0) / 8 = 0.5625 meters.
The general form of Simpson's 1/3 rule for n=8 (even number of segments) is given by:
M = (h/3) * [V(0) + 2 * ∑V(even) + 4 * ∑V(odd) + V(last)]
Here, V(even) denotes the sum of V(x) at all even-numbered segments except the first and last, and V(odd) denotes the sum of V(x) at all odd-numbered segments.
After carrying out these calculations, you will have the value of the bending moment M for the beam at a length of 4.5 m.