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Chapter 28: Problem 15

The following equation can be used to model the deflection of a sailboat mast subject to a wind force: \(\frac{d^{2} y}{d z^{2}}=\frac{f}{2 E I}(L-z)^{2}\) where \(f=\) wind force, \(E=\) modulus of elasticity, \(L=\) mast and \(I=\) moment of inertia. Calculate the deflection if \(y=0\) and \(d y / d z=0\) at \(z=0 .\) Use parameter values of \(f=60, L=30, E=\) \(1.25 \times 10^{8},\) and \(I=0.05\) for your computation.

Short Answer

Expert verified
After integrating the given equation twice and using the initial conditions, we find the deflection function to be: \[y(z) = \frac{f}{16EI}(Lz^3 - 2z^4 + z^5) + C_1z + C_2\] Plugging the given parameter values into the equation, we get: \[y(z) = \frac{60}{16(1.25 \times 10^8)(0.05)}(30z^3 - 2z^4 + z^5)\] Finally, we simplify the expression to obtain the deflection: \[y(z) = 3 \times 10^{-6}(30z^3 - 2z^4 + z^5)\]

Step by step solution

01

Integrate the given equation once

First, we will integrate the given equation with respect to \(z\) to find an equation for the first derivative \(\frac{dy}{dz}\). Integrating the equation yields: \[\frac{dy}{dz} = \int{\frac{f}{2EI}(L-z)^2 dz}\]
02

Integrate the equation again

Now we have to integrate the equation one more time with respect to \(z\) to find the actual deflection \(y(z)\). Integrating the equation yields: \[y(z) = \int\frac{dy}{dz} dz = \int\int{\frac{f}{2EI}(L-z)^2 dz} dz\]
03

Find the constants of integration

To find the constants of integration, we will use the initial conditions: \(y(0)=0\) and \(\frac{dy}{dz}(0)=0\). Let \(C_1\) and \(C_2\) be the constants of integration that we will find by applying the initial conditions.
04

Plug in the parameter values

Now we will plug in the given parameter values into our equation to find the actual deflection \(y(z)\). We are given: - \(f = 60\) - \(L = 30\) - \(E = 1.25 \times 10^8\) - \(I = 0.05\) Plug these values into the equation, and find the deflection \(y(z)\) using the constants of integration \(C_1\) and \(C_2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Deflection of Beams
The deflection of beams is a common problem in structural engineering. It refers to the movement of a beam when a force is applied to it, causing it to bend. In this context, the deflection is the displacement of the beam from its original, un-stressed position.
Understanding beam deflection is vital for ensuring the structural integrity of constructions like bridges and masts.

For a mathematical approach, the deflection can be modeled using differential equations. For example, the beam equation provided for the sailboat mast involves a second-order differential equation.
It describes how external wind forces, elasticity of the material, and moment of inertia affect the beam's bending. By solving this equation through integration, we can calculate how much the mast will bend under typical conditions.

The initial and boundary conditions are essential. For the example mast, when no deflection and no slope (derivative) of deflection are initially present, these conditions are used to solve for any unknown constants post-integration. This ensures that your final model accurately reflects the physical situation.
Numerical Integration
Numerical integration is a practical mathematical tool used when an integral cannot be solved analytically. It approximates the area under curves, which in engineering problems, like our beam deflection task, corresponds to calculating total deflection over a span.

Numerical methods like the trapezoidal rule or Simpson's rule help compute these integrals, especially when dealing with functions with no straightforward antiderivative.

In structural engineering, this approach helps solve complex equations by transforming differential equations into algebraic ones that are more manageable.
In our exercise, numerical integration supports solving the differential equation modeling the mast deflection, enabling engineers to predict how a sailboat mast will respond to wind forces effectively.
  • This involves breaking down the task into smaller, incrementally simpler calculations.
  • Each calculation provides incremental insights into the curve鈥檚 area or slope.
Boundary Value Problems
Boundary value problems (BVPs) are a type of differential equation where the solution is found by ensuring certain conditions are met at the boundaries of the domain.
These conditions make BVPs essential for defining physical systems where parameters are constant or known at the endpoints, like our sailboat mast at the base and tip.

In our scenario, BVPs help us determine the deflection since we know the mast's condition at the bottom: no initial deflection and no tilt. These known points allow us to backtrack and solve for constants during integration astronomically.
Boundary conditions transform the general solution of a differential equation into a specific physical application, ensuring solutions adhere to real-world constraints. They guide where and how forces interact with structures.

By integrating the differential equation for deflection with respect to known conditions at both ends, structural engineers predict loads and determine safe design challenges.

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Most popular questions from this chapter

Although the model in Prob. 28.18 works adequately when population growth is unlimited, it breaks down when factors such as food shortages, pollution, and lack of space inhibit growth. In such cases, the growth rate itself can be thought of as being inversely proportional to population. One model of this relationship is \(G=G^{\prime}\left(p_{\text {nlax }}-p\right)\) where \(G^{\prime}=\) a population-dependent growth rate (per people-year) and \(p_{\max }=\) the maximum sustainable population. Thus, when population is small \(\left(p \ll p_{\max }\right),\) the growth rate will be at a high constant rate of \(G^{\prime} p_{\max } .\) For such cases, growth is unlimited and Eq. (P28.19) is essentially identical to Eq. (P28.18). However, as population grows (that is, \(p\) approaches \(p_{\max }\) ), \(G\) decreases until at \(p=p_{\max }\) it is zero. Thus, the model predicts that, when the population reaches the maximum sustainable level, growth is nonexistent, and the system is at a steady state. Substituting Eq. (P28.19) into Eq. (P28.18) yields \(\frac{d p}{d t}=G^{\prime}\left(p_{\max }-p\right) p\) For the same island studied in Prob. 28.18 , employ Heun's method (without iteration) to predict the population at \(t=20\) years, using a step size of 0.5 year. Employ values of \(G=10^{-5}\) per people-year and \(p_{\max }=20,000\) people. At time \(t=0,\) the island has a population of 6000 people. Plot \(p\) yersus \(t\) and interpret the shape of the curve.

A nonisothermal batch reactor can be described by the fol- lowing equations: \(\frac{d C}{d t}=-e^{(-10 /(T+273))} C\) \(\frac{d T}{d t}=1000 e^{(-10 /(T+273))} C-10(T-20)\) where \(C\) is the concentration of the reactant and \(T\) is the temperature of the reactor. Initially the reactor is at \(15^{\circ} \mathrm{C}\) and has a concentration of reactant \(C\) of 1.0 gmol/L. Find the concentration and temperature of the reactor as a function of time.

Population-growth dynamics are important in a variety of planning studies for areas such as transportation and waterresource engineering. One of the simplest models of such growth incorporates the assumption that the rate of change of the population \(p\) is proportional to the existing population at any time \(t\) \(\frac{d p}{d t}=G p\) where \(G=\) a growth rate (per year). This model makes intuitive sense because the greater the population, the greater the number of potential parents. At time \(l=0,\) an island has a population of 6000 people. If \(G=0.075\) per year, employ Heun's method (without iteration) to predict the population at \(t=20\) years, using a step size of 0.5 year. Plot \(p\) versus \(t\) on standard and semilog graph paper. Determine the slope of the line on the semilog plot. Discuss your results.

The basic differential equation of the elastic curve for a cantilever beam (Fig. \(\mathrm{P} 28.22\) ) is given as \(E I \frac{d^{2} y}{d x^{2}}=-P(L-x)\) where \(E=\) the modulus of elasticity and \(l=\) the moment of inertia. Solve for the deflection of the beam using a numerical method. The following parameter values apply: \(E=30,000 \mathrm{ksi}, I=800 \mathrm{in}^{4}\) \(P=1\) kip, \(L=10 \mathrm{ft} .\) Compare your numerical results to the analytical solution, \(y=-\frac{P L x^{2}}{2 E I}+\frac{P x^{3}}{6 E I}\)

The temperature distribution in a tapered conical cooling fin (Fig. \(P 28.39\) ) is described by the following differential equation, which has been nondimensionalized \\[ \frac{d^{2} u}{d x^{2}}+\left(\frac{2}{x}\right)\left(\frac{d u}{d x} p u\right)=0 \\] where \(u=\) temperature \((0 \leq u \leq 1), x=\) axial distance \((0 \leq x \leq 1)\) and \(p\) is a nondimensional parameter that describes the heat transfer and geometry \\[ p=\frac{h L}{k} \sqrt{1+\frac{4}{2 m^{2}}} \\] where \(h=\) a heat transfer coefficient, \(k=\) thermal conductivity, \(L=\) the length or height of the cone, and \(m=\) the slope of the cone wall. The equation has the boundary conditions \\[ u(x=0)=0 \quad u(x=1)=1 \\] Sotye this equation for the temperature distribution using finite difference methods. Use second-order accurate finite difference analogues for the derivatives. Write a computer program to obtain the solution and plot temperature versus axial distance for various values of \(p=10,20,50,\) and 100.
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