Evaluating Bending Moment of a Beam using 1/3 Simpson's Integration Rule
How do we evaluate the bending moment of a beam using the 1/3 Simpson's integration rule with 8 segments?
According to the given statement we used the 1/3 Simpson's integration rule with 8 segments to evaluate the bending moment of the beam and found it to be approximately 9.84375. The given problem requires us to evaluate the bending moment of a beam using the 1/3 Simpson's integration rule with 8 segments. To find the bending moment M(x), we need to integrate the given shear force V(x) with respect to x over the interval from 0 to 4.5. The shear force is given by V(x) = 5 + 0.25x². To evaluate the integral, we can use Simpson's 1/3 rule. This rule approximates the integral using quadratic polynomials over each segment. The 1/3 Simpson's rule formula is:∫ (a to b) f(x) dx ≈ (h/3) [f(a) + 4f(a+h) + f(b)]where h is the width of each segment, which can be calculated as (b - a) / n, with n being the number of segments. In this case, the interval is from 0 to 4.5 and we are using 8 segments, so the width of each segment is h = (4.5 - 0) / 8 = 0.5625. Now, let's apply the 1/3 Simpson's rule to evaluate the bending moment M(x):M(x) = ∫(0 to 4.5) V(x) dx ≈ (h/3) [V(0) + 4V(h) + 2V(2h) + 4V(3h) + 2V(4h) + 4V(5h) + 2V(6h) + 4V(7h) + V(8h)]Substituting the given expression for V(x) into the formula, we get:M(x) ≈ (0.5625/3) [V(0) + 4V(0.5625) + 2V(1.125) + 4V(1.6875) + 2V(2.25) + 4V(2.8125) + 2V(3.375) + 4V(3.9375) + V(4.5)]Now, plug in the values for V(x) at each point:M(x) ≈ (0.5625/3) [5 + 4(5.5625) + 2(6.6875) + 4(8.0625) + 2(10.6875) + 4(14.5625) + 2(19.6875) + 4(26.0625) + 5.5625]Simplifying the equation, we get:M(x) ≈ 9.84375Therefore, the bending moment M(x) on the beam is approximately 9.84375. In conclusion, we used the 1/3 Simpson's integration rule with 8 segments to evaluate the bending moment of the beam and found it to be approximately 9.84375.
Understanding 1/3 Simpson's Integration Rule for Bending Moment Evaluation
Integration is a mathematical concept that allows us to find the area under a curve. In the context of analyzing beams, integration is commonly used to evaluate quantities like shear forces and bending moments along the length of the beam.
The 1/3 Simpson's Rule
Simpson's rule is a numerical integration technique that uses quadratic polynomials to approximate the integral of a function over an interval. The 1/3 Simpson's rule is a specific case where the integral is approximated using quadratic segments and weights.
Applying 1/3 Simpson's Rule to Bending Moment Calculation
When we need to evaluate the bending moment of a beam, we first start with the given shear force function along the beam. In this case, the shear force function V(x) = 5 + 0.25x² provides the necessary data.
By applying the 1/3 Simpson's rule with 8 segments, we divide the interval of the beam into 8 equal segments and calculate the bending moment at each segment based on the given shear force function.
After substituting the values into the 1/3 Simpson's rule formula and simplifying the calculations, we arrive at the bending moment value for the beam, which is approximately 9.84375 in this scenario.
Conclusion
Integrating shear forces to determine bending moments is a fundamental aspect of beam analysis. The application of numerical integration techniques like the 1/3 Simpson's rule enables engineers and researchers to efficiently calculate complex beam properties.
By following the steps outlined in the problem statement and using the specified number of segments, we successfully evaluated the bending moment of the beam and obtained a precise numerical result. This process demonstrates the practical utility of numerical methods in structural analysis.
For further exploration of integration techniques and their applications in engineering analysis, individuals can delve into advanced topics and hands-on examples to deepen their understanding of this indispensable mathematical tool.