Determine the equation of the elastic curve using the coordinate x, and specify the slope at point A and the deflection at point C. EI is constant.
The equation of elastic curve using the coordinate
Answer to Problem 7.1P
The equation of elastic curve is shown below.
The slope at point A is,
The deflection at C is,
Explanation of Solution
Calculation:
The following figure shows the free body diagram of the beam.
Figure-(1)
Write the Equilibrium Equation for the sum of horizontal forces.
Here, horizontal reaction at A is
Write the Equilibrium Equation for the sum of vertical forces.
Here, vertical reactions at point A and B are
Due to symmetry of the beam, the reactions at point A and B will be half of the total load acting on the beam.
Consider the section x-x at distance
Write the Equation for sum of moment about x-x.
Write the differential equation of the elastic curve as shown below.
Substitute
Here,
Calculate the value of
Apply the boundary conditions at the support points.
At
Substitute
Calculate the value of
At
Substitute
Calculate the slope equation.
Substitute
Calculate the deflection equation.
Substitute
Calculate the slope at A.
Substitute
Calculate the deflection at C.
Substitute
Conclusion:
The equation of elastic curve is shown below.
The slope at point A is,
The deflection at C is,
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