Uncertain Mechanics - 2

Classical Mechanics Level 3

Consider beam A C AC supported at one end A A and at another point B B as indicated on the diagram. Gravity acts downwards and the weight per unit length of the rod is a constant w w . The parameter w w is not exactly known. It is known that it is normally distributed with an expected value of μ = 30000 N / m \mu = 30000 \ N/m and a standard deviation of σ = 5000 N / m \sigma = 5000 \ N/m . It is also known that the support A A cannot withstand a reaction force greater than 8500 N 8500 \ N . If the reaction at A A exceeds the aforementioned value, then the support fails.

Compute the probability of failure of the support at A A .

Note:

  • L = 1 m L= 1 \ m

  • The weight is uniformly distributed throughout the length of the rod.

  • The supports of the rod are indicated by blue circles in the diagram.


The answer is 0.2118554.

Karan Chatrath by Karan Chatrath , 27, Netherlands

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2 solutions

Dark Angel
Jun 29, 2020

3 Helpful 2 Interesting 3 Brilliant 0 Confused

@DARK ANGEL from where you get this calculator?

A Former Brilliant Member - 11 months, 2 weeks ago
Steven Chase
Jun 29, 2020

Comments/explanation are in the attached code 

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import math

# Constants

u = 30000.0       # mean
sigma = 5000.0    # standard deviation

Amax = 8500.0     # max withstand force at A
L = 1.0           # Beam length

w1 = u - 6.0*sigma  # min value of w
w2 = u + 6.0*sigma  # max value of w

dw = (w2 - w1)/(10.0**6.0)  # w increment

##########################################

# Sweep w over a range

w = w1

num = 0.0         # Conditional integral of prob density function
denom = 0.0   # Unconditional integral of prob density function (normalizing factor)  

while w <= w2:

    W = w*L        # beam weight

    # W*(L/2.0) = B*(2.0*L/3.0)   # torque balance about A

    B = W*(L/2.0)/(2.0*L/3.0)    # force at B
    A = W - B                    # force at A

    pwr = -0.5*(((w-u)/sigma)**2.0) # gaussian exponent used to calculate PDF (prob density function)
    PDF = math.exp(pwr)*dw   # probability density function

    if A > Amax:
        num = num + PDF*dw   # If support fails, add PDF*dw to num quantity

    denom = denom + PDF*dw   # Always add PDF*dw to denom quantity 

    w = w + dw

##########################################

print (num/denom)
#0.211855977391

2 Helpful 2 Interesting 2 Brilliant 0 Confused

@Steven Chase sir how can we make 2D images in python representing 3D things.
Suggest me some website or video to learn.
Thanks in advance.

A Former Brilliant Member - 11 months, 2 weeks ago

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