L
= unsupported beam length = 400 mm
total length is 500 mm, supports 50 mm from
ends
b = beam width = 100 mm
h = beam height = 24.7 mm
E = Modulus of elasticity = 9.03x10^{10} Pa
= 9.03x10^{4} N/mm^{2
}
I = Moment of Inertia = b*h^{3}/12 =
1.26x10^{5} mm^{4}^{
}EI = (9.03x10^{4}*1.26x10^{5})
= 1.13x10^{10} N mm^{2
}Z = Section Modulus =
I/(h/2) =
1.02x10^{4}
mm^{3
}W = central load = 0.5 Kg = 4.9 N/mm^{2
}
x = point coordinate relative to support
Y(x) = deflection = (Wx/48EI)(3L^{2
}- 4x^{2})
S(x) = extreme fiber stress = -Wx/2Z
Calculate central deflection and stress: x = L/2
Ycent = Ymax = Y(L/2) = WL^{3}/48EI=
(4.90*400^{3})/(48*1.13x10^{10})
= 580 nm
See Beam Test Data 2 in good agreement
Scent = Smax = -WL/4Z= -(4.90*400)/(4*1.02x10^{4})
= .048 N/mm^{2} = 5x10^{3}
Pa
The Schott web site
lists bending strengths for bonded Zerodur pieces in the range of 25
to 50 MPa; an older Schott publication listed Zerodur bending
strengths of 70 to 120 MPa depending on surface finish.
Tilt = T = dY/dx = d/dx[(W/48EI)(3xL^{2}-4x^{3})]
= [W/16EI](L^{2} -
4x^{2})
with max at supports, minimum at center
Tmax = T(0) = WL^{2}/16EI
= (4.90*400^{2})/(16*1.13x10^{10})=
4.3 microradians
Tcent = T(L/2) = 0
See
Beam Test Data 1, in good agreement. |