Flexure engineering fundamental: Leaf Spring

Construction Fundamentals

S-shape stiffness

$C_x=\frac{Ebt^3}{L^3} $
$C_y=\frac{Etb^3}{L^3} $
$C_z=\frac{EA}{L}$ 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

$F_{z\ buckling}=4\frac{\pi^2EI}{L^2}$ only if $u_x=0$
$\sigma_{max}=\frac{M_{max}}{I}\frac{1}{2}t=\frac{\frac{F_xL}{2}}{I}\frac{1}{2}t=\frac{F_xLt}{4I}$

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

Leaf Spring S-Shape Deformation

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

$F_{z\ buckling}=2.045\frac{\pi^2EI}{L^2}$ only if $u_x=0$

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

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

Leaf Spring C-Shape

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