Symbolic operations in MAPLE

Question

I need assistance on how to achieve the following manipulation in Maple Software.

The questions are how do I get eq. 3.1b from 3.1a, 3.3 from 3.2a and 3.9b from 3.9a?

In 3.1b the integration was delayed only the partial differentiation took place.

In 3.3 only the internal integral of 3.2a was executed and curl brackets being the integration wrt x, while prime and dot represent differentiation wrt x and z respectively.

In 3.9b nearly the same operations occurred from 3.9a.

I will be grateful if I get hints on how to go about the operations as shown in the equations below.

Kind regards.

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$$W\left(z\right) = q_s L \sin\left(\frac{\pi z}{L}\right) \tag{2.1a}\label{2.1a} $$ $$\theta\left(z\right) = q_t \pi \cos \left(\frac{\pi z}{L}\right) \tag{2.1b}\label{2.1b}$$ $$\gamma_{xz,o} = \frac{dW}{dz}-\theta=\left(q_{s}-q_{t}\right)\pi \cos \left(\frac{\pi z}{L}\right) \tag{2.2}\label{2.2}$$ $$u\left(x,z\right) = -\frac{2 x}{b} u\left(x\right) \tag{2.3a}\label{2.3a}$$ $$w_f\left(x,z\right) = f\left(x\right) w_f\left(z\right) \tag{2.3b}\label{2.3b}$$ $$w_w\left(y,z\right) = g\left(y\right) w_w\left(z\right)\tag{2.3c}\label{2.3c}$$ $$f\left(x\right) = -\frac{4 x}{b} + \left(\frac{2 x}{b}\right)^2 + \left(\frac{2 x}{b}\right)^3 + \sin\left(\frac{\pi x}{b}\right) \tag{2.4}\label{2.4}$$ $$g\left(y\right) = \sin\left(\frac{\pi y}{h}\right) \tag{2.5}\label{2.5}$$

\n

$$U_{B}=\frac{1}{2}EI_{w}\int_{0}^L \left(\frac{\partial^2 W}{\partial z^2}\right)^2 dz \tag{3.1a}\label{3.1a} $$ $$=\frac{1}{2}EI_{w}\int_{0}^L q_{s}^2 \frac{\pi^4}{L^2} \sin^2 \left(\frac{\pi z}{L}\right) dz \tag{3.1b}\label{3.1b}$$

$$U_{bf}=\frac{D_{f}}{2}\int_{0}^L\int_{-b/2}^0 \left \{ \left(\frac{\partial^2 w_{f}}{\partial x^2} + \frac{\partial^2 w_{f}}{\partial z^2}\right)^2-2\left(1-\nu\right)\left[\frac{\partial^2 w_{f}}{\partial x^2}\frac{\partial^2 w_{f}}{\partial z^2}-(\frac{\partial^2 w_{f}}{\partial x \partial z})^2\right] \right\} dxdz\tag{3.2a}\label{3.2a}$$

$$U_{bw}=\frac{D_{w}}{2}\int_{0}^L\int_{-h}^0 \left \{ \left(\frac{\partial^2 w_{w}}{\partial y^2} + \frac{\partial^2 w_{w}}{\partial z^2}\right)^2-2\left(1-\nu\right)\left[\frac{\partial^2 w_{w}}{\partial y^2}\frac{\partial^2 w_{w}}{\partial z^2}-(\frac{\partial^2 w_{w}}{\partial y \partial z})^2\right] \right\} dxdz\tag{3.2b}\label{3.2b}$$

$$$$ $$U_{bf}=\frac{D_{f}}{2}\int_{0}^L [\{f^{\prime\prime2}\}_{x} w_{f}^2+\{f^2\}_{x} \ddot w_{f}^2 +2\nu\{f^{\prime\prime} f\}_{x} \ddot w_{f}w_{f} +2\left(1-\nu\right)\{f^{\prime2}\}_{x} \dot w_{f}^2 ] dz\tag{3.3}\label{3.3}$$

$$U_{bw}=\frac{D_{w}}{2}\int_{0}^L [\{g^{\prime\prime2}\}_{y} w_{w}^2+\{g^2\}_{y} \ddot w_{w}^2 +2\nu\{g^{\prime\prime} g\}_{y} \ddot w_{w}w_{w} +2\left(1-\nu\right)\{g^{\prime2}\}_{y} \dot w_{w}^2 ] dz\tag{3.4}\label{3.4}$$

$$ \{f^{\prime\prime} f\}_{x} = \int_{-b/2}^0\left\{\left[\frac{d^2}{dx^{2}}f\left(x\right)\right]f\left(x\right)\right\}dx \tag{3.5}\label{3.5} $$

$$ \{g^{\prime\prime} g\}_{y} = \int_{-h}^0\left\{\left[\frac{d^2}{dx^{2}}g\left(y\right)\right]g\left(y\right)\right\}dy \tag{3.6}\label{3.6}$$

$$\varepsilon_{zft}=\frac{\partial u_{t}}{\partial z}-\Delta \tag{3.7}\label{3.7}$$

$$\varepsilon_{zfc}=\frac{\partial u_{t}}{\partial z}-\Delta +\frac{\partial u}{\partial z} +\frac{1}{2}\left(\frac{\partial w_{f}}{\partial z}\right)^2 \tag{3.8}\label{3.8}$$ where $u_{t}=-\theta x$

$$U_{mf}=\frac{E t_{f}}{2}\int_{0}^L\int_{-b/2}^0 \varepsilon_{zft}^2 dxdz + \frac{E t_{f}}{2}\int_{0}^L\int_{-b/2}^0 \varepsilon_{zfc}^2 dxdz \tag{3.9a}\label{3.9a}$$

$$ =\frac{E t_{f} b}{2}\int_{0}^L\left[\frac{b^2\pi^4}{12L^2} q_{t}^2 \sin^2 \left(\frac{\pi z}{L}\right)+\frac{\dot u^2}{6}+\frac{\{f^4\}_{x}}{4b} \dot w_{f}^4-q_{t}\frac{b\pi^2}{6L}\sin\left(\frac{\pi z}{L}\right)\dot u-\frac{\Delta}{2}\dot u+q_{t}\frac{\pi^2 \{x f^2\}_{x}}{bL}\sin\left(\frac{\pi z}{L}\right)\dot w_{f}^2-\frac{\{f^2\}_{x}\Delta}{b}\dot w_{f}^2-\frac{2\{xf^2\}_{x}}{b^2}\dot u\dot w_{f}^2+\Delta^2 \right] \tag{3.9b}\label{3.9b}$$

{Mode interaction of global and local buckling in thin-walled I-section struts with rigid flange-web joints}

Abdul.

Answer 1

Using the Maple command PDEtools:-declare we can suppress the argument and have W(z) display as just W, etc. And that supports prime (but not dot) notation for ordinary derivatives.

Alternatively the Typesetting:-Settings command allows for both prime or dot notation, but that way does not allow for suppression of the arguments in the function calls.

For your first question, you only need consider utilizing equations 2.1a and the result from mathematical substitution.

restart;
PDEtools:-declare(W(z));

eq2_1a := W(z) = q[s]*L*sin(Pi*z/L):

U__B := 1/2*EI[w]*Int(Diff(W(z),z,z)^2,z=0..L);

eval(U__B,eq2_1a);

For your second question you need to utilize equation 2.3b.

Here that is, using the PDEtools:-declare mechanism that I described.

restart;
PDEtools:-declare(w__f(x,z));
PDEtools:-declare(w__f(z));
PDEtools:-declare(f(x));
PDEtools:-declare(prime=x);
eqn2_3b := w__f2 = proc(x,z) f(x)*w__f(z); end proc:
U__bf := int((diff(w__f2(x,z),x,x)+diff(w__f2(x,z),z,z))^2
             -2*(1-v)*(diff(w__f2(x,z),x,x)*diff(w__f2(x,z),z,z)
                       -diff(w__f2(x,z),x,z)^2),
             x=-b/2..0);
eval(U__bf, [eqn2_3b]);
simplify(%);

And here is it using the Typesetting:-Settings mechanism that I described.

restart;
Typesetting:-Settings(prime=x):
Typesetting:-Settings(typesetprime=true):
Typesetting:-Settings(dot=z):
Typesetting:-Settings(typesetdot=true):

eqn2_3b := w__f2 = proc(x,z) f(x)*w__f(z); end proc:

U__bf := int((diff(w__f2(x,z),x,x)+diff(w__f2(x,z),z,z))^2
             -2*(1-v)*(diff(w__f2(x,z),x,x)*diff(w__f2(x,z),z,z)
                       -diff(w__f2(x,z),x,z)^2),
             x=-b/2..0);

eval(U__bf, [eqn2_3b]);
simplify(%);

I will not attempt to reproduce the final notation of using subscripted squiggly braces to denote the inner integration. That doesn't even show the integration bounds, and is inconsistent with the earlier use of such brackets to denote mere grouping of terms.