Description

E = 2.1000e+11
I = (1/12)*10^(-8)
I = 8.3333e-10
q0 = 100
q0 = 100
L = 0.1
L = 0.1000
F = 40
F = 40
c.
% n = 1
displayFormula(“q11 = int(q0*x, x, 0, L)”)

q11 = int(q0*x, x, 0, L)
q11 =

displayFormula(“q12 = int((-q0/L)*x^2 + 2*q0*x^1, x, L, 2*L)”)

q12 = int((-q0/L)*x^2 + 2*q0*x^1, x, L, 2*L)
q12 =

q1 = q11 + q12
q1 =

F1 = q1 – 2*F*L – (7/6)*q0*L^2 + 2*F*L
F1 =
% n = 2
displayFormula(“q21 = int(q0*x^2, x, 0, L)”)

q21 = int(q0*x^2, x, 0, L)
q21 =

displayFormula(“q22 = int((-q0/L)*x^3 + 2*q0*x^2, x, L, 2*L)”)

q22 = int((-q0/L)*x^3 + 2*q0*x^2, x, L, 2*L)
q22 =

q2 = q21 + q22
q2 =

F2 = q2 – F*(2*L)^2

A = (E*I) * [0 0; 0 int(4, x, 0, 2*L)]
A =

B = [F1; F2]
B =

X = double(linsolve(A, B))
X = 2×1
0
-0.0105
% n = 3
displayFormula(“q31 = int(q0*x^3, x, 0, L)”)

q31 = int(q0*x^3, x, 0, L)
q31 =

displayFormula(“q32 = int((-q0/L)*x^4 + 2*q0*x^3, x, L, 2*L)”)

q32 = int((-q0/L)*x^4 + 2*q0*x^3, x, L, 2*L)
q32 =

q3 = q31 + q32
q3 =

F3 = q3 – F*(2*L)^3
F3 =

A = (E*I) * [int(4, x, 0, 2*L) int(12*x, x, 0, 2*L); int(12*x, x, 0, 2*L) int(36*x^2, x, 0, 2*L)]
A =

B = [F2; F3]
B =

X = double(linsolve(A,B))
X = 2×1
-0.0204
0.0329
% n = 4
displayFormula(“q41 = int(q0*x^4, x, 0, L)”)

q41 = int(q0*x^4, x, 0, L)
q41 =

displayFormula(“q42 = int((-q0/L)*x^5 + 2*q0*x^4, x, L, 2*L)”)

q42 = int((-q0/L)*x^5 + 2*q0*x^4, x, L, 2*L)
q42 =

q4 = q41 + q42
q4 =

F4 = q4 – F*(2*L)^4
F4 =

A = (E*I) * [int(4, x, 0, 2*L) int(12*x, x, 0, 2*L) int(24*x^2, x, 0, 2*L); int(12*x, x, 0, 2*L) int(36*x^2, x, 0, 2*L) int(72*x^3, x, 0, 2*L); int(24*x^2, x, 0, 2*L) int(72*x^3, x, 0, 2*L) int(144*x^4, x, 0, 2*L)]
A =

B = [F2; F3; F4]
B =

X = double(linsolve(A,B))
X = 3×1
-0.0196
0.0248
0.0201
% n = 5
displayFormula(“q51 = int(q0*x^5, x, 0, L)”)

q51 = int(q0*x^5, x, 0, L)
q51 =

displayFormula(“q52 = int((-q0/L)*x^6 + 2*q0*x^5, x, L, 2*L)”)

q52 = int((-q0/L)*x^6 + 2*q0*x^5, x, L, 2*L)
q52 =

q5 = q51 + q52
q5 =

F5 = q5 – F*(2*L)^5
F5 =

A = (E*I) * [int(4, x, 0, 2*L) int(12*x, x, 0, 2*L) int(24*x^2, x, 0,
2*L) int(40*x^3, x, 0, 2*L);
int(12*x, x, 0, 2*L) int(36*x^2, x, 0, 2*L) int(72*x^3, x, 0,
2*L) int(120*x^4, x, 0, 2*L);
int(24*x^2, x, 0, 2*L) int(72*x^3, x, 0, 2*L) int(144*x^4, x, 0,
2*L) int(240*x^5, x, 0, 2*L);
int(40*x^3, x, 0, 2*L) int(120*x^4, x, 0, 2*L) int(240*x^5, x, 0, 2*L) int(400*x^6, x, 0, 2*L)]
A =

B = [F2; F3; F4; F5]
B =

X = double(linsolve(A,B))
X = 4×1
-0.0195
0.0229 0.0320
-0.0238
% plotting
fplot((-0.0105)*x^2, [0, 0.2]) hold on fplot((-0.0204)*x^2 + (0.0329)*x^3, [0, 0.2]) fplot((-0.0196)*x^2 + (0.0248)*x^3 + (0.0201)*x^4, [0, 0.2]) fplot((-0.0195)*x^2 + (0.0229)*x^3 + (0.0320)*x^4 + (-0.0238)*x^5, [0, 0.2]) hold off
xlabel(‘x’) ylabel(‘w(x)’) legend(‘n=2’, ‘n=3’, ‘n=4’, ‘n=5’)

% checking derivative diff(diff(x^2))*diff(diff(x^2))
ans =
diff(diff(x^2))*diff(diff(x^3))
ans =
diff(diff(x^2))*diff(diff(x^4))
ans =
diff(diff(x^2))*diff(diff(x^5))
ans =
diff(diff(x^3))*diff(diff(x^2))
ans =
diff(diff(x^3))*diff(diff(x^3))
ans =
diff(diff(x^3))*diff(diff(x^4))
ans =
diff(diff(x^3))*diff(diff(x^5))
ans =
diff(diff(x^4))*diff(diff(x^2))
ans =
diff(diff(x^4))*diff(diff(x^3))
ans =
diff(diff(x^4))*diff(diff(x^4))
ans =
diff(diff(x^4))*diff(diff(x^5))
ans =
diff(diff(x^5))*diff(diff(x^2))
ans =
diff(diff(x^5))*diff(diff(x^3))
ans =
diff(diff(x^5))*diff(diff(x^4))
ans =
diff(diff(x^5))*diff(diff(x^5))
ans =
d.
% n = 2 syms a1 a2 u_h u_h = a1*x + a2*x^2
u_h =
I2 = 0.5*E*I*int(diff(diff(u_h))^2, x, 0, 2*L) – int(u_h*q0, x, 0, L) – int(u_h*((q0/L)*x+2*q0), x, L, 2*L) + F*(a1*2*L+a2*((2*L)^2))
I2 =

diff(I2, a2)
ans =

% n = 3 syms a1 a2 a3 u_h u_h = a1*x + a2*x^2 + a3*x^3
u_h =
I3 = 0.5*E*I*int(diff(diff(u_h))^2, x, 0, 2*L) – int(u_h*q0, x, 0, L) – int(u_h*((q0/L)*x+2*q0), x, L, 2*L) + F*(a1*2*L + a2*((2*L)^2) + a3*((2*L)^3))
I3 =

diff(I3, a2)
ans =

diff(I3, a3)
ans =

A = [140 42; 42 84/5]
A = 2×2
140.0000 42.0000
42.0000 16.8000
B = [-59/40; -609/2000]
B = 2×1
-1.4750
-0.3045
X = double(linsolve(A,B))
X = 2×1
-0.0204
0.0329
% n = 4 syms a1 a2 a3 a4 u_h
u_h = a1*x + a2*x^2 + a3*x^3 + a4*x^4
u_h =
I4 = 0.5*E*I*int(diff(diff(u_h))^2, x, 0, 2*L) – int(u_h*q0, x, 0, L) – int(u_h*((q0/L)*x+2*q0), x, L, 2*L) + F*(a1*2*L + a2*((2*L)^2) + a3*((2*L)^3) + a4*((2*L)^4))
I4 =

diff(I4, a2)
ans =

diff(I4, a3)
ans =

diff(I4, a4)
ans =

A = [140 42 56/5; 42 84/5 126/25; 56/5 126/25 1008/625]
A = 3×3
140.0000 42.0000 11.2000
42.0000 16.8000 5.0400
11.2000 5.0400 1.6128
B = [-59/40; -609/2000; -619/10000]
B = 3×1
-1.4750
-0.3045
-0.0619
X = double(linsolve(A,B))
X = 3×1
-0.0196
0.0248
0.0201
% n = 5
syms a1 a2 a3 a4 a5 u_h
u_h = a1*x + a2*x^2 + a3*x^3 + a4*x^4 + a5*x^5
u_h =
I5 = 0.5*E*I*int(diff(diff(u_h))^2, x, 0, 2*L) – int(u_h*q0, x, 0, L) – int(u_h*((q0/L)*x+2*q0), x, L, 2*L) + F*(a1*2*L + a2*((2*L)^2) + a3*((2*L)^3) + a4*((2*L)^4) + a5*((2*L)^5))
I5 =

diff(I5, a2)
ans =

diff(I5, a3)
ans =

diff(I5, a4)
ans =

diff(I5, a5)
ans =

A = [140 42 56/5 14/5; 42 84/5 126/25 168/125; 56/5 126/25 1008/625 56/125; 14/5
168/125 56/125 16/125]
A = 4×4
140.0000 42.0000 11.2000 2.8000
42.0000 16.8000 5.0400 1.3440 11.2000 5.0400 1.6128 0.4480
2.8000 1.3440 0.4480 0.1280
B = [-59/40; -609/2000; -619/10000; -5249/420000]
B = 4×1
-1.4750
-0.3045
-0.0619
-0.0125
X = double(linsolve(A,B))
X = 4×1
-0.0195
0.0229 0.0320
-0.0238
f.
syms u2 u3 u4 u5 u2 = -0.0105*x^2
u2 =

u3 = -0.0204*x^2 + 0.0329*x^3
u3 =

u4 = -0.0196*x^2 + 0.0248*x^3 + 0.0201*x^4
u4 =

u5 = -0.0195*x^2 + 0.0229*x^3 + 0.0320*x^4 – 0.0238*x^5
u5 =

% n=2 u25 = u2 – u5
u25 =

I25 = double(0.5*E*I*int(diff(diff(u25))^2, x, 0, 2*L) – int(u25*q0, x, 0, L) – int(u25*((-q0/L)*x+2*q0), x, L, 2*L) + F*((-0.0105*(2*L)^2) – (-0.0195*(2*L)^2 + 0.0229*(2*L)^3 + 0.0320*(2*L)^4 – 0.0238*(2*L)^5)))
I25 = 0.0069
% n=3 u35 = u3 – u5
u35 =

I35 = double(0.5*E*I*int(diff(diff(u35))^2, x, 0, 2*L) – int(u35*q0, x, 0, L)
– int(u35*((-q0/L)*x+2*q0), x, L, 2*L) + F*((-0.0204*(2*L)^2 + 0.0329*(2*L)^3) –
(-0.0195*(2*L)^2 + 0.0229*(2*L)^3 + 0.0320*(2*L)^4 – 0.0238*(2*L)^5)))
I35 = 4.3308e-05
% n=4 u45 = u4 – u5
u45 =

I45 = double(0.5*E*I*int(diff(diff(u45))^2, x, 0, 2*L) – int(u45*q0, x, 0, L)
– int(u45*((-q0/L)*x+2*q0), x, L, 2*L) + F*((-0.0196*(2*L)^2 + 0.0248*(2*L)^3 + 0.0201*(2*L)^4) – (-0.0195*(2*L)^2 + 0.0229*(2*L)^3 + 0.0320*(2*L)^4 –
0.0238*(2*L)^5)))
I45 = -8.0235e-06