PROBLEM 12.3
Determine the elastic line equation for the beam using the x coordinate valid for 0 (smallest equal) x (smallest L/2). Specify the slope at A and the maximum deflection of the beam. EI is constant.
00:00 Interpretation of the Problem and Calculation of Reactions in Supports
✅ Note that as our load P is applied in the middle of the beam, therefore, for its equilibrium, the reaction at each support has to be P/2.
01:40 Determining the Function of the Bending Moment of the Beam
✅ In this step of the problem we have to apply the section method, that is, make a "cut" at a distance x from the support A of the beam. In this small "piece" of beam you will represent all the loads acting on the beam, such as the reaction at support A and the internal bending moment to be determined.
By summing the moments in relation to the point where the "cut" was made (at a distance x from A) it is possible to determine the function of the bending moment.
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02:30 Integrating the Bending Moment Function
✅ Once we have determined the moment function, now we have to integrate this function once to get the beam slope equation and by integrating this equation again we will find the elastic line equation.
05:00 Boundary Conditions and the Elastic Line Equation
✅ The purpose of applying the boundary conditions is to find the integration constants C1 and C2 that appeared in the slope and elastic line equations after the integration. And how to apply such conditions?
Note that in a simply supported beam with a load applied in the middle of the beam, the displacement will be zero at its supports and the slope of the beam will be zero at the middle of the beam (L/2). Thus, at x = 0: v = 0, by substituting this first condition in the elastic line equation, we easily find the value of the constant C2.
C1 will be determined by applying the second boundary condition to the slope equation, that is, at x = L/2: dv/dx = 0.
Now we just substitute the constant values ​​in the elastic line equation to get the equation as a function of flexural stiffness EI.
07:45 Calculation of A-Slope and Maximum Beam Deflection
✅ Very well, to determine the slope on support A (ϴA), we have to substitute the correct value of x in the slope equation, that is, the slope will be obtained for x = 0.
We know that the maximum deflection (vmax) occurs in the middle of the beam, so substitute the value of x = L/2 into the elastic line equation.
I hope you enjoyed the personal content, thanks in advance and good studies!
📚 Source: Mechanics of Materials 7th Ed. (R.C. Hibbeler)
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