Direct integration of a beam

Direct integration is a <a href="/facts/Structural_analysis/phRJdfup">structural analysis</a> method for measuring internal shear, internal moment, rotation, and deflection of a beam.
For a beam with an applied weight 
 
 
 
 w
 (
 x
 )
 
 
 {\displaystyle w(x)}
 
, taking downward to be positive, the internal <a href="/facts/Shear_force/hVKPYRar">shear force</a> is given by taking the negative integral of the weight:

V
        (
        x
        )
        =
        −
        ∫
        w
        (
        x
        )
        
        d
        x
      
    
    {\displaystyle V(x)=-\int w(x)\,dx}

The internal <a href="/facts/Bending_moment/6tmafWtR">moment</a> 
 
 
 
 M
 (
 x
 )
 
 
 {\displaystyle M(x)}
 
 is the integral of the internal shear:

M
        (
        x
        )
        =
        ∫
        V
        (
        x
        )
        
        d
        x
      
    
    {\displaystyle M(x)=\int V(x)\,dx}
  
 = 
  
    
      
        −
        ∫
        
          [
          
            ∫
            w
            (
            x
            )
            
            d
            x
          
          ]
        
        d
        x
      
    
    {\displaystyle -\int \left[\int w(x)\,dx\right]dx}

The <a href="/facts/Angle_of_rotation/a4VjkXGe">angle of rotation</a> from the horizontal, 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
, is the integral of the internal moment divided by the product of the <a href="/facts/Young%2527s_modulus/MV5JMCIM">Young's modulus</a> and the <a href="/facts/Second_moment_of_area/TludHUKK">area moment of inertia</a>:

θ
        (
        x
        )
        =
        
          
            1
            
              E
              I
            
          
        
        ∫
        M
        (
        x
        )
        
        d
        x
      
    
    {\displaystyle \theta (x)={\frac {1}{EI}}\int M(x)\,dx}

Integrating the angle of rotation obtains the vertical displacement 
 
 
 
 ν
 
 
 {\displaystyle \nu }
 
:

ν
        (
        x
        )
        =
        ∫
        θ
        (
        x
        )
        
        d
        x
      
    
    {\displaystyle \nu (x)=\int \theta (x)\,dx}

Direct integration of a beam open-in-new

Direct integration of a beam