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Question: Solving an Euler-Bernoulli beam with uniform load and axial varying force

Question:Solving an Euler-Bernoulli beam with uniform load and axial varying force

OffshoreEngineer 25 Maple 16
dsolve
July 18 2013
1

Hi all,

This might be more a mathematical problem than a Maple problem, but still maybe somebody could help me out.

I am trying to solve the differential equation of an Euler-Bernoulli beam with an distributed load (q1) and a varying axial force (N(x)). The following equation applies:

 

 

I would like to solve this for a simply supported beam. Below I have a Maple file where I am trying to solve it. The general solution gives me airy functions and when I insert the BC's; no solution at all is found. I would like to find a complete solution for w1.



``

> 

N := proc (x) options operator, arrow; 1-(1/100)*x end proc

 

proc (x) options operator, arrow; 1-(1/100)*x end proc

 (1)

 

> 

deqns := EL*IN*(diff(w1(x), x, x, x, x))-(diff(N(x)*(diff(w1(x), x)), x)) = -q1;

 

EL*IN*(diff(diff(diff(diff(w1(x), x), x), x), x))+(diff(w1(x), x))/L-(1-x/L)*(diff(diff(w1(x), x), x)) = -q1

 (2)

 

> 

dsolve(deqns);

 

w1(x) = Int((AiryAi((1/(L*EL*IN))^(1/3)*(L-x))*_C3*EL*IN*(1/(L*EL*IN))^(1/3)+AiryBi((1/(L*EL*IN))^(1/3)*(L-x))*_C2*EL*IN*(1/(L*EL*IN))^(1/3)+Pi*(Int(AiryBi((1/(L*EL*IN))^(1/3)*(L-x))*(_C1*EL*IN-q1*x), x))*AiryAi((1/(L*EL*IN))^(1/3)*(L-x))-Pi*(Int(AiryAi((1/(L*EL*IN))^(1/3)*(L-x))*(_C1*EL*IN-q1*x), x))*AiryBi((1/(L*EL*IN))^(1/3)*(L-x)))/(EL*IN*(1/(L*EL*IN))^(1/3)), x)+_C4

 (3)

 

> 

inits := w1(0) = 0, ((D@@2)(w1))(0) = 0, w1(100) = 0, ((D@@2)(w1))(100) = 0;

 

w1(0) = 0, ((D@@2)(w1))(0) = 0, w1(100) = 0, ((D@@2)(w1))(100) = 0

 (4)

 

> 

totsys := {deqns, inits};

 

{EL*IN*(diff(diff(diff(diff(w1(x), x), x), x), x))+(diff(w1(x), x))/L-(1-x/L)*(diff(diff(w1(x), x), x)) = -q1, w1(0) = 0, w1(100) = 0, ((D@@2)(w1))(0) = 0, ((D@@2)(w1))(100) = 0}

 (5)

 

> 

dsolve(totsys);

 

> 

``

 

> 

``

 



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