An 11m beam is subjected to a load, and the shear force follows the equation V(x)=5+0.25x^2 where V is the shear force, x is the length in distance along the beam, V=dM/dx where M is the bending moment M=Mo+integration of Vdx from 0 to x. If Mo is zero and x=11, calculate M using MULTIPLE SEGMENT TRAPEZOIDAL RULE. use 2.2m increments
Integrate 5 + 0.25x^2, from x=0 to x = 11, using the trapezoidal rule of numerical integration.
To do it the way they want, compute V(x) at x = 0, x = 2.2 m , x = 4.4 m, x = 6.6 m, x = 8.8 m and x = 11 m.
I assume you know how the trapezoidal rule works. If not, look it up in your class notes, text or online.
Basically, you add up the V(x) values at the four interior points, add the average of V(x=0) and V(x=11), and multiply the sum by the increment width, 2.2 m.
To calculate the bending moment using the multiple segment trapezoidal rule, we need to divide the length of the beam into smaller segments of equal width, in this case, 2.2m increments.
Let's calculate the bending moment for each segment step-by-step:
1. First, calculate the initial shear force at x = 0. Plug in x = 0 into the equation V(x) = 5 + 0.25x^2:
V(0) = 5 + 0.25(0)^2 = 5
2. Now, we can calculate the bending moment at x = 2.2m. Since the beam is divided into 2.2m increments, we need to calculate the area under the shear force curve from x = 0 to x = 2.2m using the trapezoidal rule.
The formula for the area under the curve using the trapezoidal rule is:
ΔM = Δx * (V(x1) + V(x2)) / 2
Δx = 2.2m
V(x1) = 5 (shear force at x = 0)
V(x2) = V(2.2) = 5 + 0.25(2.2)^2 = 9.05
ΔM = 2.2 * (5 + 9.05) / 2 = 20.05
3. Repeat the above steps for each segment until x = 11m, using the previous shear force value for each subsequent segment.
Segment 2 (x = 4.4m):
ΔM = 2.2 * (V(2.2) + V(4.4)) / 2
V(4.4) = V(2.2 + 2.2) = V(4.4) = 5 + 0.25(4.4)^2 = 25.02
ΔM = 2.2 * (9.05 + 25.02) / 2 = 52.81
Segment 3 (x = 6.6m):
ΔM = 2.2 * (V(4.4) + V(6.6)) / 2
V(6.6) = V(4.4 + 2.2) = V(6.6) = 5 + 0.25(6.6)^2 = 44.61
ΔM = 2.2 * (25.02 + 44.61) / 2 = 97.69
Segment 4 (x = 8.8m):
ΔM = 2.2 * (V(6.6) + V(8.8)) / 2
V(8.8) = V(6.6 + 2.2) = V(8.8) = 5 + 0.25(8.8)^2 = 70.58
ΔM = 2.2 * (44.61 + 70.58) / 2 = 154.38
Segment 5 (x = 11m):
ΔM = 2.2 * (V(8.8) + V(11)) / 2
V(11) = V(8.8 + 2.2) = V(11) = 5 + 0.25(11)^2 = 109.75
ΔM = 2.2 * (70.58 + 109.75) / 2 = 209.27
4. Now, sum up all the calculated ΔM values to get the total bending moment at x = 11m:
M = ∑ ΔM
M = 20.05 + 52.81 + 97.69 + 154.38 + 209.27 = 534.2
Therefore, the bending moment at x = 11m is 534.2.
To calculate the bending moment (M) using the Multiple Segment Trapezoidal Rule, we need to break down the beam into smaller segments and approximate the area under the shear force curve (V) for each segment.
Given:
Length of the beam (x) = 11m
Shear force equation (V(x)) = 5 + 0.25x^2
Increment size (Δx) = 2.2m
Step 1: Divide the beam into smaller segments.
Since the increment size is 2.2m, we can divide the beam into segments of length 2.2m each. We'll have a total of 5 segments: [0, 2.2], [2.2, 4.4], [4.4, 6.6], [6.6, 8.8], and [8.8, 11].
Step 2: Approximate the area for each segment.
To approximate the area under the curve in each segment, we'll use the trapezoidal rule. The formula for each segment is: Area = (V1 + V2) * Δx / 2, where V1 and V2 are the shear forces at the two ends of the segment, and Δx is the length of the segment.
For the first segment [0, 2.2]:
V1 = V(0) = 5 + 0.25(0)^2 = 5
V2 = V(2.2) = 5 + 0.25(2.2)^2 = 6.105
Area1 = (5 + 6.105) * 2.2 / 2 = 11.7275
Similarly, calculate the areas for the remaining segments:
For the second segment [2.2, 4.4]:
V1 = V(2.2) = 6.105
V2 = V(4.4) = 5 + 0.25(4.4)^2 = 12.46
Area2 = (6.105 + 12.46) * 2.2 / 2 = 19.627
For the third segment [4.4, 6.6]:
V1 = V(4.4) = 12.46
V2 = V(6.6) = 5 + 0.25(6.6)^2 = 23.49
Area3 = (12.46 + 23.49) * 2.2 / 2 = 38.709
For the fourth segment [6.6, 8.8]:
V1 = V(6.6) = 23.49
V2 = V(8.8) = 5 + 0.25(8.8)^2 = 41.44
Area4 = (23.49 + 41.44) * 2.2 / 2 = 86.383
For the fifth segment [8.8, 11]:
V1 = V(8.8) = 41.44
V2 = V(11) = 5 + 0.25(11)^2 = 67.75
Area5 = (41.44 + 67.75) * 2.2 / 2 = 167.148
Step 3: Sum up the areas of all segments.
M = Area1 + Area2 + Area3 + Area4 + Area5
= 11.7275 + 19.627 + 38.709 + 86.383 + 167.148
= 323.5945
Therefore, using the Multiple Segment Trapezoidal Rule with 2.2m increments, the bending moment (M) at x = 11 is approximately 323.5945 units.