Last Updated on September 23, 2026 by Maged kamel
Maximum Deflection distance for a supported Beam under a triangular load-Numerically.
We want to determine the maximum Deflection distance for a triangular Load acting on a supported Beam 6.0 m long. The maximum Load intensity is 2.0 kN/m. Our task is to find the x-max distance using the Numerical method. We will solve this using the Newton-Raphson method to find the roots.
The three equations for the change in slope & slope, and Deflection values.
This is a List of three items: the first is the change-of-curvature equation, and the diagram shows a supported Beam under a triangular Load, where w0 is the maximum Value of the triangular Load, and x is the distance for which we want to calculate the change of curvature.
The second item is the curvature slope equation, and the diagram shape for a supported Beam under a triangular Load, where P is the maximum Value of the triangular Load, and x is the distance for which we want to calculate the curvature slope.
The third item shows the slope and curvature diagram for a supported Beam under a triangular Load, where w0 is the maximum Value of the triangular Load and x is the distance for which we want to calculate the Deflection.
The next slide image shows the slope and Deflection equations of a supported Beam under a uniformly Triangularly distributed Load.
We need to estimate the maximum Deflection distance from the left support, x max. The Point of maximum Deflection is where the slope of the curvature is zero. Usually, we use the Newton-Raphson method; we want to find the Point where y = 0.
The function Value f(x) is in the numerator. Refer to the relation, where X0 is the starting pPoint f(x0) is the function Value at x0, and f'(x0) is the function’s slope Value at Point X0.
The revised Newton-Raphson equation is based on the slope equation.
To use the Newton-Raphson equation, we will work with the y’ curve, since it changes from positive to negative. A root Point for zero slopes will give us the Point of maximum Deflection. Instead of dealing with the y-curve as in the original Newton-Raphson equation, we now deal with the function.
The steps for estimating x max using the analytical solution.
The modified equation for the root Point will be as follows, for the first Point after setting an initial Point, which will be given a Value of 2.00m, less than L/2.
Our first trial Value of x0 is = 2.00 m.
We need f'(2) and f ” (2) to apply them in the adjusted equation. We have the Expression for f'(x), the slope of the curve due to the triangular Load, and f”(x), the change in curvature of the Beam. We plug in x0 = 2.0 m and get the two values of f'(2) and ” f ” (2), as shown in the next slide. We need to use 1/EI=1/8*10^4.
Our second trial Value of x0.
Then we plug in x0 as a second trial = 3.30 m, and we get the second maximum Deflection distance of 3.1169 m. Our third x0, starting at 3.1169 m, yields a maximum Deflection distance of 3.1169 m.
Then we plug in x0 as a second trial (X2 = 3.1169 m) and obtain the third maximum Deflection distance of 3.116 m.
The following Table shows the result of x0 when we start with an initial Point of 2.0m for four iterations.
This graph shows the slope. We start at x0 = 2 m and end up at x1 = 3.30 m. Please refer to the previous Table.
If our first trial Value of x0 is 3.00m.
Suppose somebody wishes to choose a starting Point of 3.00 m. x0 = 3.0m leads to a modified Value of 3.1166m as the maximum Deflection distance.
Then, we plug in x0 as a second trial Value of 3.1166 m and obtain the second maximum Deflection distance of 3.116 m.
The following Table shows the result of x0 when we start with an initial Point of 3.0m for four iterations.
Suppose somebody wishes to choose a starting Point as=4.0. x0 with a starting Value of 4.00m leads to a modified Value of 3.116m as the maximum Deflection distance.
The Modified Newton-Raphson equation.
We can use the modified Newton-Raphson Equation to determine the Point of maximum Deflection, but we will consider the following terms: f(x) from the slope equation, F'(x) from the curvature slope, F ” (x) from the Shear equation, and Q(x). The following slide reminds us of the equations needed to obtain the values for Load, Shear, and Moment. We use the discontinuity function expressions.
We need to estimate the values of Shear and Moment for a starting Point x0 = 2.00 m, as we can see Q(x0=2) = 1.66 kN, while the Value of M(X) for x0=2.0 m equals 3.555 KN · m.
We obtain the slope at x0 = 2.0 m, which is f(x) in the modified Newton-Raphson Expression. The Value of EIV dash, or the slope, equals -4.662 kN/m222. We plug the Modified Newton-Raphson equation into the equation and get x1 = 2.8739 m. Please refer to the following slide for more details
The following slide shows the details of estimating the different values of Xi using a starting Point of X0 = 2.00 m. Thanks a lot.
You can review or download the PDF file used for this Post from the following documents.
This is a Link to the previous Post, where I included a solved Problem demonstrating the application of the Newton-Raphson equations for structural analysis.-structural analysis numerically by the Newton-Raphson method.
This is a good external reference—a holistic numerical method.
This Link is useful for a numerical analysis calculator.