Topic: Flat Plate & Maximum Principal Stress Theory
Subject: Machine Design
Tool: QCAD & Scilab
- Circular Flat Plate Application
This topic will present the solution for the design of an acrylic material used as a cover in the vacuum chamber. Figure 1 illustrates an application wherein the thickness of circular plate cover is calculated for the considerable factor-of-safety in order to overcome the induced stress from a vacuum pressure.
Figure 1. Vacuum Chamber with Circular Plate Cover Application.
For a circular plate under uniform load and simply supported at the edge, the maximum stress at the center is
$$\sigma = \frac{3\,(3 + \nu)\,p \,r^2 }{8\,t^2} \tag{1}$$
- Maximum Principal Stress Theory
This failure theory is accredited to W.J.M. Rankine. It gives good predictions and often used for brittle materials. It presumes that when the maximum principal stress exceeds a certain limiting value, for this case the ultimate strength, \(\sigma_u\), for brittle materials failure occurs. This design stress for uniaxial load with shear is given by the Equation (2)
$$\sigma_d = \frac{\sigma}{2} + \sqrt{\left( \frac{\sigma}{2} \right)^2 + \tau^2} \tag{2}$$
The normal stress from Equation (2) is the maximum bending stress, \(\sigma\), as in this case of circular plate cover. Shear stress, \(\tau\), is calculated from the force due to the difference between atmospheric pressure and vacuum acting on the area around the chamber opening circumference with the given thickness of the cover plate. Shear stress is defined as
$$\tau = \frac{F}{A}, \quad F = \pi\,r^2\,p, \quad A = 2\,\pi\,r\,t \tag{3}$$
For a static loading of a brittle material the limiting stress is taken as the ultimate strength and the design stress is expressed as
$$\sigma_d = \frac{\sigma_u}{N} \tag{4}$$
Our objective for the design is to determine the thickness of the circular plate cover which will give a considerable factor-of-safety while addressing other factors such as stress concentration due to the effect of attaching vacuum fittings or other devices and the availability of material based on standard or off-the-shelf thickness.
- Scilab Script
// Copyright (C) 2025 - Gani Comia // Date of creation: 2 Apr 2025 // Circular Plate, Uniform Load, Simply Supported Edge clear;clc; // module or functions // (1) maximum stress at the center due to pressure function sigma_fp=flatPlate(p, r, t, nu) // calculate the stress on the flat plate. // input argument: // p - MPa or N/mm^2, uniform load // r - mm, radius of circular plate // t - mm, plate thickness // nu - Poisson's ratio // output argument: // sigma_fp - MPa, maximum principal stress at the center sigma_fp = (3*(3+nu).*p.*r.^2)./(8*t.^2); endfunction // (2) maximu shear stress at the chamber opening function tau_fp=shearPlate(p, r, t) // calculate the shear at the plate around the chamber opening. // input argument: // p - MPa or N/mm^2, uniform load // r - mm, radius of circular plate // t - mm, plate thickness // output argument: // tau_fp - MPa, maximum shear stress c = 2*%pi*r; // mm, circumference a_shear = c.*t; // mm^2, area under shear a_normal = %pi*r.^2; // mm^2, area perpendicular to pressure F = p.*a_normal; // N, force due to pressue tau_fp = F./a_shear; endfunction // (3) maximum principal stress theory function sigma_d=max_Pst(sigma, tau) // calculate the design stress based on max principal stress theory // input argument: // sigma - MPa, principal stress (uniaxial) // tau - MPa, shear stress // output argument: // sigma_d - MPa, design stress based on MPST sigma_d = (sigma/2)+sqrt((sigma/2).^2 + tau.^2); endfunction // (4) factor of safety function N=FoS(sigma_d, sigma_u) // calculate the factor of safety for brittle material // input argument: // sigma_d - MPa, design stress // sigma_u - MPa, ultimate strength // output argument: // N - no unit, factor-of-safety N = sigma_u./sigma_d; endfunction // main function // (5) primary parameters p_atm = 760; // Torr, atmospheric pressure p_vac = 0.001; // Torr, vacuum pressure r = 500/2; // mm, radius of flat circular cover nu = 0.35; // Poisson's ratio of acrylic or // polymethyl methacrylate of PMMA sigma_u = 75; // MPa, ultimate tensile strength of PMMA // (6) secondaty parameters p_eqv = p_atm-p_vac; // Torr, equivalent pressue p_eqv = p_eqv*0.000133322; // MPa, equivalent pressure // (7) thickness domain under examination t = linspace(1,75,200) // (8) calculation sigma_fp = flatPlate(p_eqv,r,t,nu) tau_fp = shearPlate(p_eqv,r,t) sigma_d = max_Pst(sigma_fp,tau_fp) N = FoS(sigma_d,sigma_u) // (9) visualization of results clf; fig = gcf() fig.figure_size = [700,700]; plot(t,N,"b-","linewidth",3.5) title("Thickness of Vacuum Chamber Cover","fontsize",4.5) xlabel("Design Thickness (t), mm","fontsize",4) ylabel("Factor-of-Safety (N)","fontsize",4) xgrid(3,0) note="$\LARGE N = f(t),\quad\text{with Max Principal Stress Theory}$" legend(note,with_box = %F) xstring(50,53,"https://gani-mech-toolbox.blogspot.com") // (10) plot of standard thickness and factor-of-safety std_thk = [25 50 75] Nstdthk = length(std_thk) format(5) for i = 1:Nstdthk N(i) = interp1(t,N,std_thk(i)) vLx = [std_thk(i) std_thk(i)]; vLy = [0 N(i)]; hLx = [0 std_thk(i)]; hLy = [N(i) N(i)]; plot(vLx,vLy,"r--") plot(hLx,hLy,"r--") plot(std_thk(i),N(i),"marker","o","markerFaceColor","red") xstring(5,N(i),["$\large N = $",string(N(i))]) xstring(std_thk(i)+3.5,0,["$\large t = $",string(std_thk(i))],-90) end ax = gca() ax.data_bounds = [0 0; 80 60];
- Scilab Output (Figure 2)
Figure 2. Factor-of-Safety as a function of thickness for the design of vacuum chamber cover.
A vacuum chamber cover with the thickness of \(\text{50 mm}\) would be reasonable enough for the consideration such that its weight will provide an initial compression set for gasket in between the chamber and the cover, but a toggle clamp or any clamping mechanism is still necessary for a leak free sealing, and the provision of fittings for inlet/outlet vacuum port and instruments at the chamber cover leads to an increase in stress concentration that in effect reduce the factor-of-safety.
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References
- M.F. Spotts. Design of Machine Elements. 6th Ed. Englewood Cliffs, N.J. 07632. Prentice-Hall, Inc. 1985.
- Virgil Moring Faires. Design of Machine Elements. 4th Ed. The Macmillan Company, New York. 1968.
