|
Parameter |
Equation |
|
Reaction Force 1 [R1] |
$${ R }_{ 1 }=\frac { { w }_{ a } }{ 2{ L }^{ 3 } } { (L-a) }^{ 3 }(L+a)+\frac {
{ w }_{ L }-{ w }_{ a } }{ 20{ L }^{ 3 } } { (L-a) }^{ 3 }(3L+2a)$$ |
|
Reaction Force 2 [R2] |
$${ R }_{ 2 }=\frac { { w }_{ a }+{ w }_{ L } }{ 2 } \cdot (L-a)-{ R }_{ 1 }$$ |
|
Shear force at distance x [V] |
$$V={ R }_{ 1 }-{ w }_{ a }\left< x-a \right> -\frac { { w }_{ L }-{ w }_{ a }
}{ 2\cdot (L-a) } { \left< x-a \right> }^{ 2 }$$ |
|
Reaction Moment 1 [M1] |
$${ M }_{ 1 }=-\frac { { w }_{ a } }{ 12{ L }^{ 2 } } { (L-a) }^{ 3
}(L+3a)-\frac { { w }_{ L }-{ w }_{ a } }{ 60{ L }^{ 2 } } { (L-a) }^{ 3
}(2L+3a)$$ |
|
Reaction Moment 2 [M2] |
$${ M }_{ 2 }={ R }_{ 1 }L+{ M }_{ 1 }-\frac { { w }_{ a } }{ 2 } { (L-a) }^{ 2
}-\frac { { w }_{ L }-{ w }_{ a } }{ 6 } { (L-a) }^{ 2 }$$ |
|
Moment at distance x
[M] |
$$M={ M }_{ 1 }+{ R }_{ 1 }x-\frac { { w }_{ a } }{ 2 } { \left< x-a \right> }^{
2 }-\frac { { w }_{ L }-{ w }_{ a } }{ 6(L-a) } { \left< x-a \right> }^{ 3 }$$ |
|
Bending stress at distance x [σ] |
$$\sigma =\frac { Mc }{ I } $$ |
|
End Deflection 1 [y1] |
$${ y }_{ 1 }=0$$ |
|
End Deflection 2 [y2] |
$${ y }_{ 2 }=0$$ |
|
Deflection at distance x [y] |
$$y={ y }_{ 1 }+{ \theta }_{ 1 }x+\frac { { M }_{ 1 }{ x }^{ 2 } }{ 2EI } +\frac
{ { R }_{ 1 }{ x }^{ 3 } }{ 6EI } -\frac { { w }_{ a } }{ 24EI } { \left< x-a
\right> }^{ 4 }-\frac { { w }_{ L }-{ w }_{ a } }{ 120EI(L-a) } { \left< x-a
\right> }^{ 5 }$$ |
|
Slope 1 [θ1] |
$${ \theta }_{ 1 }=0$$ |
|
Slope 2 [θ2] |
$${ \theta }_{ 2 }=0$$ |
|
Slope [θ] |
$$\theta ={ \theta }_{ 1 }+\frac { { M }_{ 1 }x }{ EI } +\frac { { R }_{ 1 }{ x
}^{ 2 } }{ 2EI } -\frac { { w }_{ a } }{ 6EI } { \left< x-a \right> }^{ 3
}-\frac { { w }_{ L }-{ w }_{ a } }{ 24EI(L-a) } { \left< x-a \right> }^{ 4 }$$ |
|
Parameter |
Equation |
|
Reaction Force 1 [R1] |
$${ R }_{ 1 }=\frac { P }{ { L }^{ 3 } } { (L-a) }^{ 2 }(L+2a)$$ |
|
Reaction Force 2 [R2] |
$${ R }_{ 2 }=\frac { P{ a }^{ 2 } }{ { L }^{ 3 } } (3L-2a)$$ |
|
Shear force at distance x [V] |
$$VM={ R }_{ 1 }-P{ \left< x-a \right> }^{ 0 }$$ |
|
Reaction Moment 1 [M1] |
$${ M }_{ 1 }=\frac { -Pa }{ { L }^{ 2 } } { (L-a) }^{ 2 }$$ |
|
Reaction Moment 2 [M2] |
$${ M }_{ 2 }=\frac { -P{ a }^{ 2 } }{ { L }^{ 2 } } { (L-a) }$$ |
|
Moment at distance x
[M] |
$$M={ M }_{ 1 }+{ R }_{ 1 }x-P\left< x-a \right>$$ |
|
Bending stress at distance x [σ] |
$$\sigma =\frac { Mc }{ I } $$ |
|
End Deflection 1 [y1] |
$${ y }_{ 1 }=0$$ |
|
End Deflection 2 [y2] |
$${ y }_{ 2 }=0$$ |
|
Deflection at distance x [y] |
$$y={ y }_{ 1 }+{ \theta }_{ 1 }x+\frac { { M }_{ 1 }{ x }^{ 2 } }{ 2EI } +\frac
{ { R }_{ 1 }{ x }^{ 3 } }{ 6EI } -\frac { P }{ 6EI } { \left< x-a \right> }^{ 3
}$$ |
|
Slope 1 [θ1] |
$${ \theta }_{ 1 }=0$$ |
|
Slope 2 [θ2] |
$${ \theta }_{ 2 }=0$$ |
|
Slope [θ] |
$$\theta ={ \theta }_{ 1 }+\frac { { M }_{ 1 }{ x } }{ EI } +\frac { { R }_{ 1
}{ x }^{ 2 } }{ 2EI } -\frac { P }{ 2EI } { \left< x-a \right> }^{ 2 }$$ |
Note: In these formulas, equations in brackets "< >" are singularity
functions.
|
Parameter |
Equation |
|
Reaction Force 1 [R1] |
$${ R }_{ 1 }=-\frac { 6{ M }_{ o }a }{ { L }^{ 3 } } (L-a)$$ |
|
Reaction Force 2 [R2] |
$${ R }_{ 1 }=\frac { 6{ M }_{ o }a }{ { L }^{ 3 } } (L-a)$$ |
|
Shear force at distance x [V] |
$$V={ R }_{ 1 }$$ |
|
Reaction Moment 1 [M1] |
$${ M }_{ 1 }=-\frac { { M }_{ o } }{ { L }^{ 2 } } ({ L }^{ 2 }-4aL+3{ a }^{ 2
})$$ |
|
Reaction Moment 2 [M2] |
$${ M }_{ 2 }=\frac { { M }_{ o } }{ { L }^{ 2 } } (3{ a }^{ 2 }-2aL)$$ |
|
Moment at distance x
[M] |
$$M={ M }_{ 1 }+{ R }_{ 1 }x+{ M }_{ o }{ \left< x-a \right> }^{ 0 }$$ |
|
Bending stress at distance x [σ] |
$$\sigma =\frac { Mc }{ I } $$ |
|
End Deflection 1 [y1] |
$${ y }_{ 1 }=0$$ |
|
End Deflection 2 [y2] |
$${ y }_{ 2 }=0$$ |
|
Deflection at distance x [y] |
$$y={ y }_{ 1 }+{ \theta }_{ 1 }x+\frac { { M }_{ 1 }{ x }^{ 2 } }{ 2EI } +\frac
{ { R }_{ 1 }{ x }^{ 3 } }{ 6EI } +\frac { { M }_{ o } }{ 2EI } { \left< x-a
\right> }^{ 2 }$$ |
|
Slope 1 [θ1] |
$${ \theta }_{ 1 }=0$$ |
|
Slope 2 [θ2] |
$${ \theta }_{ 2 }=0$$ |
|
Slope [θ] |
$$\theta ={ \theta }_{ 1 }+\frac { { M }_{ 1 }{ x } }{ EI } +\frac { { R }_{ 1
}{ x }^{ 2 } }{ 2EI } +\frac { { M }_{ o } }{ EI } { \left< x-a \right> }$$ |
Note: In these formulas, equations in brackets "< >" are singularity
functions.