using a symbolic function of a parabolic arch, find the tension, shear, and moment forces, given an applied vertical point force at any position in matlab

To find the tension, shear, and moment forces in a parabolic arch given an applied vertical point force at any position, you can use symbolic functions in Matlab.

First, you need to define the symbolic variables for the position of the point force and the variables for the arch geometry. Let's assume the position of the applied force is represented by x, and the variables for the arch geometry are the span length L, the arch height h, and the arch base width b.

main.m
syms x L h b
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Next, you can define the symbolic function that represents the parabolic shape of the arch. The equation of a parabola in vertex form is given by y = a*(x - h)^2 + k, where (h, k) is the vertex of the parabola and a determines the width. In the case of a symmetrical parabolic arch, the vertex is at the center of the span (i.e., (L/2, h)).

main.m
y = h - (4*h/L^2) * (x-L/2)^2;
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To find the tension force at any position x, you can take the derivative of the parabolic equation with respect to x.

main.m
tension = diff(y, x);
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To find the shear force, you can integrate the tension force equation with respect to x.

main.m
shear = int(tension, x);
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And finally, to find the moment force, you can integrate the shear force equation with respect to x.

main.m
moment = int(shear, x);
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Now you can substitute the desired value of x into each of the equations to calculate the corresponding tension, shear, and moment forces.

main.m
x = ...; % specify the value of x
tension_at_x = subs(tension, x);
shear_at_x = subs(shear, x);
moment_at_x = subs(moment, x);
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Remember to replace ... with the desired value of x.

Note: This code assumes that the arch is symmetrical and follows a parabolic shape. If your particular arch has a different shape or if you need to consider additional factors, you may need to modify the equations accordingly.