CodeCogs

Beam deflection due to a point load with two pin jointed supports at either end.

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Interface

#include <codecogs/engineering/structures/deflection_s.h>

using namespace Engineering::Structures;

double deflection_s(double L, double a, double W, double x, double EI=1)

This function is calculates the deflection $\delta(x)$ along a cantilever beam that is fixed solidly at one end and subjected to a single perpendicular point load W at a position a from the fixed end.

The deflection of the beam is calculated using the principal of virtual work, applying a unit load of 1N at the location where the deflection is required and solving

$\delta = \frac{1}{EI} \int_0^L M(x) M_u(x) dx$

(1)

where M(x) are the moment along the beam due to the applied unit load W, while M_u(x) is the moment due to the applied virtual point load at the location x.

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Example 1

Calculate the deflection along a 6m long solid steel beam (10mm x 5mm) due to a person that weights 35kg who stands mid way along it length.

In graphical form the deflection (negative is downwards) is given by: \graph L=9 a=3 x=0:9 W=-686 EI=10e9 .size=medium

#include<stdio.h>
#include<codecogs/engineering/structures/deflection_pp.h>

int main()
{
  double E=10e9;      // approximate strength of wood.
  double I=1.042e-6;  // I=b*h^3/12.
  double L=9;
  double a=3;
  double W=70*9.81; // Force in Newtons
  for(int x=0;x<=L;x++)
    printf("\n deflection(x=%d)=%lf",x,
      Engineering::Structures::deflection_pp(L, a, W, x, E*I));
  return 0;
}

Output:

deflection(x=0)=0.000000
deflection(x=1)=0.322188
deflection(x=2)=0.600441
deflection(x=3)=0.790825
deflection(x=4)=0.860389
deflection(x=5)=0.820115
deflection(x=6)=0.691972
deflection(x=7)=0.497927
deflection(x=8)=0.259947
deflection(x=9)=0.000000

Parameters

The length of the beam. [m]

The location of the point load applied to the beam. [m]

The point load applied at location a in a direction perpendicular to the main beam. [N]

The point at which the deflection should be calculated

The Modulus of Elasticity (E) multiplied with the Second Moment of Area (I) for the specified beam. See Young's Modulus for example values. [N/m^2]

Interactive Calculator

Result –