deflection_s — Structures

Deflection of a cantilever beam due to a point load.

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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 principle 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 its length.

In graphical form the deflection (negative is downwards) is given by: \graph L=6 a=3 x=0:6 W=-343.35 EI=20.83 .size=medium

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

int main()
{
  double E=200e9;     // approximate Young's Modulus of steel.
  double I=1.0417e-10;// I=b*h^3/12, for a 10mm x 5mm cross section.
  double L=6;
  double a=3;
  double W=35*9.81; // Force in Newtons
  for(int x=0;x<=L;x++)
    printf("\n deflection(x=%d)=%lf",x,
      Engineering::Structures::deflection_s(L, a, W, x, E*I));
  return 0;
}

Output:

deflection(x=0)=0.000000
 deflection(x=1)=21.973697
 deflection(x=2)=76.907939
 deflection(x=3)=148.322454
 deflection(x=4)=222.483681
 deflection(x=5)=296.644907
 deflection(x=6)=370.806134

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 –