# Flexure engineering fundamental: Leaf spring

## Construction Fundamentals

#### S-shape stiffness

$C_x=\frac{Ebt^3}{L^3}$
$C_y=\frac{Etb^3}{L^3}$ if $R_x$ is fixed, $C_y=\frac{Etb^3}{4L^3}$ if $R_x$ is free
$C_z=\frac{EA}{L}$ only if $u_x=0$
$C_z=\frac{700EAI_y}{L(700I_y+{u_x}^2A)}$ for $u_x\neq0$

$K_x=\frac{Etb^3}{12L}$ for $u_x=0$
$K_y=\frac{Ebt^3}{12L}$ for $u_x=0$
$K_z=\frac{Gbt^3}{3L}=\frac{Ebt^3}{6\left(1+\nu\right)L}$ for $u_x=0$

#### S-shape motion characteristics

$u_x=\frac{F_x}{C_x}$
$u_{xmax}=\frac{1}{3}\frac{L^2}{Et}\sigma_{max}$
$u_z=\frac{3}{5}\frac{u_x^2}{L}$

#### S-shape force limits

$\sigma_{max}=\frac{M_{max}}{I}\frac{1}{2}t=\frac{\frac{F_xL}{2}}{I}\frac{1}{2}t=\frac{F_xLt}{4I}$

dynamic movements: $\sigma_{max}<$ fatigue stress limit
static deformation: $\sigma_{max}<$ yield stress limit $(\sigma_{0.2})$

See Beam Theory: Buckling for equations to calculate the maximum buckling load.

#### C-shape stiffness

$C_x=\frac{Ebt^3}{{4L}^3}$
$C_y=\frac{Etb^3}{4L^3}$
$C_z=\frac{EA}{L}$ only if $u_x=0$

$K_x=\frac{Etb^3}{12L}$ for $u_x=0$
$K_y=\frac{Ebt^3}{12L}$ for $u_x=0$
$K_z=\frac{Gbt^3}{3L}=\frac{Ebt^3}{6\left(1+\nu\right)L}$ for $u_x=0$

#### C-shape motion characteristics

$u_x=\frac{F_x}{C_x}$
$u_{xmax}=\frac{2}{3}\frac{L^2}{Et}\sigma_{max}$
$u_z=\frac{3}{5}\frac{u_x^2}{L}$

#### C-shape force limits

$\sigma_{max}=\frac{M_{max}}{I}\frac{1}{2}t=\frac{F_xL}{I}\frac{1}{2}t=\frac{F_xLt}{2I}$

dynamic movements: $\sigma_{max}<$ fatigue stress limit
static deformation: $\sigma_{max}<$ yield stress limit $(\sigma_{0.2})$

See Beam Theory: Buckling for equations to calculate the maximum buckling load.

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