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					<b>Critical Buckling Pressure of layered Orthotropic Cylinders with Stiffened ends under External Hydrostatic Pressure</b>
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					<b>10/12/12</b>
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					<f family="TTE1B308C0t00" charset="0">The basis of calculation for the various loadings is derived from </f>
					<f family="TTE1ECA240t00" charset="0">NASA SP-8007</f>
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					<f family="TTE1B308C0t00" charset="0">(Ref. 11.4). In addition, the interpretation of </f>
					<f family="TTE1ECA240t00" charset="0">NASA SP-8007 </f>
					<f family="TTE1B308C0t00" charset="0">found in </f>
					<f family="TTE1ECA240t00" charset="0">The</f>
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					<f family="TTE1ECA240t00" charset="0">Behavior of Structures Composed ofComposite Materials </f>
					<f family="TTE1B308C0t00" charset="0">was referenced</f>
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					<f family="TTE1B308C0t00" charset="0">(Ref. 9.1, pp. 161-169.)</f>
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					<b>REQUIRED:<sp count="5"/>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Find the Critcal Buckling Pressure for a given external hydrostatic pressure.<sp count="2"/>
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					<b>ASSUME:</b>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">1.<sp count="2"/>All lamina (layers) are the same material.<sp count="2"/>
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					<b>GIVEN:<sp count="2"/>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Cylinder Radius (inside ??):<sp count="9"/>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Lamina thickness:<sp count="3"/>
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					<b>SOLUTION:</b>
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						<ml:apply>
							<ml:mult/>
							<ml:real>8.5</ml:real>
							<ml:apply>
								<ml:pow/>
								<ml:real>10</ml:real>
								<ml:real>5</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:id xml:space="preserve">psi</ml:id>
					</ml:apply>
				</ml:define>
			</math>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">θ</ml:id>
						<ml:real>1</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:real>80</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="14"/>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">θ</ml:id>
						<ml:real>2</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:real>-80</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
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				</ml:define>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">θ</ml:id>
						<ml:real>3</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:real>80</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">θ</ml:id>
						<ml:real>4</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:real>-80</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
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				</ml:define>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">θ</ml:id>
						<ml:real>5</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:real>80</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
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				</ml:define>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">θ</ml:id>
						<ml:real>6</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:real>-80</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="19"/>
		</region>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">θ</ml:id>
						<ml:real>7</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:real>80</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="20"/>
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		<region region-id="470" left="108" top="476.4" width="67.8" height="15" align-x="127.2" align-y="486" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">θ</ml:id>
						<ml:real>8</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:real>-80</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="21"/>
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		<region region-id="471" left="186" top="476.4" width="61.2" height="15" align-x="205.2" align-y="486" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">θ</ml:id>
						<ml:real>9</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:real>80</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="22"/>
		</region>
		<region region-id="472" left="264" top="476.4" width="71.4" height="15" align-x="286.8" align-y="486" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve">θ</ml:id>
						<ml:real>10</ml:real>
					</ml:apply>
					<ml:apply>
						<ml:mult/>
						<ml:real>-80</ml:real>
						<ml:id xml:space="preserve">deg</ml:id>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="23"/>
		</region>
		<region region-id="307" left="12" top="505.8" width="183.6" height="13.2" align-x="28.8" align-y="516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Stiffness Matrix Quantities:<sp count="3"/>
				</p>
			</text>
		</region>
		<region region-id="309" left="12" top="525.6" width="96" height="34.8" align-x="39" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" subscript="11">Q</ml:id>
					<ml:apply>
						<ml:div/>
						<ml:id xml:space="preserve" subscript="11">E</ml:id>
						<ml:apply>
							<ml:minus/>
							<ml:real>1</ml:real>
							<ml:apply>
								<ml:mult/>
								<ml:id xml:space="preserve" subscript="12">ν</ml:id>
								<ml:id xml:space="preserve" subscript="21">ν</ml:id>
							</ml:apply>
						</ml:apply>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="24"/>
		</region>
		<region region-id="310" left="144" top="525.6" width="96" height="34.8" align-x="171" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" subscript="12">Q</ml:id>
					<ml:apply>
						<ml:div/>
						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve" subscript="12">ν</ml:id>
							<ml:id xml:space="preserve" subscript="22">E</ml:id>
						</ml:apply>
						<ml:apply>
							<ml:minus/>
							<ml:real>1</ml:real>
							<ml:apply>
								<ml:mult/>
								<ml:id xml:space="preserve" subscript="12">ν</ml:id>
								<ml:id xml:space="preserve" subscript="21">ν</ml:id>
							</ml:apply>
						</ml:apply>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="25"/>
		</region>
		<region region-id="312" left="264" top="525.6" width="96" height="34.8" align-x="291" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" subscript="22">Q</ml:id>
					<ml:apply>
						<ml:div/>
						<ml:id xml:space="preserve" subscript="22">E</ml:id>
						<ml:apply>
							<ml:minus/>
							<ml:real>1</ml:real>
							<ml:apply>
								<ml:mult/>
								<ml:id xml:space="preserve" subscript="12">ν</ml:id>
								<ml:id xml:space="preserve" subscript="21">ν</ml:id>
							</ml:apply>
						</ml:apply>
					</ml:apply>
				</ml:define>
			</math>
			<rendering item-idref="26"/>
		</region>
		<region region-id="313" left="378" top="536.4" width="55.2" height="16.2" align-x="405" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:id xml:space="preserve" subscript="66">Q</ml:id>
					<ml:id xml:space="preserve" subscript="12">G</ml:id>
				</ml:define>
			</math>
			<rendering item-idref="27"/>
		</region>
		<region region-id="300" left="18" top="577.8" width="130.8" height="13.2" align-x="18" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Direction Transformations:<sp count="4"/>
				</p>
			</text>
		</region>
		<region region-id="302" left="162" top="577.8" width="39" height="13.2" align-x="162" align-y="588" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<text use-page-width="false" push-down="false" lock-width="true">
				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Where:<sp count="2"/>
				</p>
			</text>
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			<math optimize="false" disable-calc="false">
				<ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30">
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						<ml:indexer/>
						<ml:id xml:space="preserve">m</ml:id>
						<ml:id xml:space="preserve">k</ml:id>
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					<ml:apply>
						<ml:id xml:space="preserve">cos</ml:id>
						<ml:apply>
							<ml:indexer/>
							<ml:id xml:space="preserve">θ</ml:id>
							<ml:id xml:space="preserve">k</ml:id>
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			</math>
			<rendering item-idref="28"/>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
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						<ml:id xml:space="preserve">n</ml:id>
						<ml:id xml:space="preserve">k</ml:id>
					</ml:apply>
					<ml:apply>
						<ml:id xml:space="preserve">sin</ml:id>
						<ml:apply>
							<ml:indexer/>
							<ml:id xml:space="preserve">θ</ml:id>
							<ml:id xml:space="preserve">k</ml:id>
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			<rendering item-idref="29"/>
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		<region region-id="301" left="18" top="604.8" width="321.6" height="24.6" align-x="66.6" align-y="618" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve" subscript="11">QBar</ml:id>
						<ml:id xml:space="preserve">k</ml:id>
					</ml:apply>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:plus/>
							<ml:apply>
								<ml:mult/>
								<ml:id xml:space="preserve" subscript="11">Q</ml:id>
								<ml:apply>
									<ml:pow/>
									<ml:parens>
										<ml:apply>
											<ml:indexer/>
											<ml:id xml:space="preserve">m</ml:id>
											<ml:id xml:space="preserve">k</ml:id>
										</ml:apply>
									</ml:parens>
									<ml:real>4</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:mult/>
								<ml:apply>
									<ml:mult/>
									<ml:apply>
										<ml:mult/>
										<ml:real>2</ml:real>
										<ml:parens>
											<ml:apply>
												<ml:plus/>
												<ml:id xml:space="preserve" subscript="12">Q</ml:id>
												<ml:apply>
													<ml:mult/>
													<ml:real>2</ml:real>
													<ml:id xml:space="preserve" subscript="66">Q</ml:id>
												</ml:apply>
											</ml:apply>
										</ml:parens>
									</ml:apply>
									<ml:apply>
										<ml:pow/>
										<ml:parens>
											<ml:apply>
												<ml:indexer/>
												<ml:id xml:space="preserve">m</ml:id>
												<ml:id xml:space="preserve">k</ml:id>
											</ml:apply>
										</ml:parens>
										<ml:real>2</ml:real>
									</ml:apply>
								</ml:apply>
								<ml:apply>
									<ml:pow/>
									<ml:parens>
										<ml:apply>
											<ml:indexer/>
											<ml:id xml:space="preserve">n</ml:id>
											<ml:id xml:space="preserve">k</ml:id>
										</ml:apply>
									</ml:parens>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve" subscript="22">Q</ml:id>
							<ml:apply>
								<ml:pow/>
								<ml:parens>
									<ml:apply>
										<ml:indexer/>
										<ml:id xml:space="preserve">n</ml:id>
										<ml:id xml:space="preserve">k</ml:id>
									</ml:apply>
								</ml:parens>
								<ml:real>4</ml:real>
							</ml:apply>
						</ml:apply>
					</ml:apply>
				</ml:define>
			</math>
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			<math optimize="false" disable-calc="false">
				<ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve" subscript="11">QBar</ml:id>
						<ml:id xml:space="preserve">k</ml:id>
					</ml:apply>
					<ml:unitOverride>
						<ml:id xml:space="preserve">psi</ml:id>
					</ml:unitOverride>
					<result xmlns="http://schemas.mathsoft.com/math30">
						<unitedValue>
							<ml:matrix rows="10" cols="1">
								<ml:real>1461514.3266618743</ml:real>
								<ml:real>1461514.3266618743</ml:real>
								<ml:real>1461514.3266618743</ml:real>
								<ml:real>1461514.3266618743</ml:real>
								<ml:real>1461514.3266618743</ml:real>
								<ml:real>1461514.3266618743</ml:real>
								<ml:real>1461514.3266618743</ml:real>
								<ml:real>1461514.3266618743</ml:real>
								<ml:real>1461514.3266618743</ml:real>
								<ml:real>1461514.3266618743</ml:real>
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							<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
								<unitReference unit="pounds_per_sq_inch"/>
							</unitMonomial>
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			<rendering item-idref="31"/>
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			<math optimize="false" disable-calc="false">
				<ml:define xmlns:ml="http://schemas.mathsoft.com/math30">
					<ml:apply>
						<ml:indexer/>
						<ml:id xml:space="preserve" subscript="12">QBar</ml:id>
						<ml:id xml:space="preserve">k</ml:id>
					</ml:apply>
					<ml:apply>
						<ml:plus/>
						<ml:apply>
							<ml:mult/>
							<ml:apply>
								<ml:mult/>
								<ml:parens>
									<ml:apply>
										<ml:minus/>
										<ml:apply>
											<ml:plus/>
											<ml:id xml:space="preserve" subscript="11">Q</ml:id>
											<ml:id xml:space="preserve" subscript="22">Q</ml:id>
										</ml:apply>
										<ml:apply>
											<ml:mult/>
											<ml:real>4</ml:real>
											<ml:id xml:space="preserve" subscript="66">Q</ml:id>
										</ml:apply>
									</ml:apply>
								</ml:parens>
								<ml:apply>
									<ml:pow/>
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										<ml:apply>
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											<ml:id xml:space="preserve">m</ml:id>
											<ml:id xml:space="preserve">k</ml:id>
										</ml:apply>
									</ml:parens>
									<ml:real>2</ml:real>
								</ml:apply>
							</ml:apply>
							<ml:apply>
								<ml:pow/>
								<ml:parens>
									<ml:apply>
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										<ml:id xml:space="preserve">n</ml:id>
										<ml:id xml:space="preserve">k</ml:id>
									</ml:apply>
								</ml:parens>
								<ml:real>2</ml:real>
							</ml:apply>
						</ml:apply>
						<ml:apply>
							<ml:mult/>
							<ml:id xml:space="preserve" subscript="12">Q</ml:id>
							<ml:parens>
								<ml:apply>
									<ml:plus/>
									<ml:apply>
										<ml:pow/>
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												<ml:apply>
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													<ml:id xml:space="preserve">k</ml:id>
												</ml:apply>
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											<ml:real>2</ml:real>
										</ml:apply>
									</ml:apply>
								</ml:parens>
								<ml:real>2</ml:real>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Shell Thickness:<sp count="3"/>
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			</text>
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								<ml:real>1387828.585101906</ml:real>
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								<ml:real>1387828.585101906</ml:real>
								<ml:real>1387828.585101906</ml:real>
								<ml:real>1387828.585101906</ml:real>
								<ml:real>1387828.585101906</ml:real>
								<ml:real>1387828.585101906</ml:real>
								<ml:real>1387828.585101906</ml:real>
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								<ml:real>1387828.585101906</ml:real>
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					<general precision="3" show-trailing-zeros="true" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mean Radius of Shell (Rn):<sp count="2"/>
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			<math optimize="false" disable-calc="false">
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Lamina distances from Mean Radius<sp count="2"/>(h):<sp count="2"/>
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							<ml:real>1</ml:real>
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			</math>
			<rendering item-idref="40"/>
		</region>
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								<ml:apply>
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									<ml:real>1</ml:real>
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						</ml:apply>
						<ml:apply>
							<ml:div/>
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							<ml:real>2</ml:real>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Elastic Constants (for stiffened multilayered Orthotropic Cylinders):<sp count="5"/>
				</p>
			</text>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">(A11) </p>
			</text>
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			<math optimize="false" disable-calc="false">
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							<ml:id xml:space="preserve">N</ml:id>
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								<ml:real>-0.061999999999999979</ml:real>
								<ml:real>-0.030999999999999989</ml:real>
								<ml:real>0</ml:real>
								<ml:real>0.031000000000000024</ml:real>
								<ml:real>0.062000000000000013</ml:real>
								<ml:real>0.093000000000000013</ml:real>
								<ml:real>0.124</ml:real>
								<ml:real>0.155</ml:real>
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											<ml:id xml:space="preserve">N</ml:id>
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								<ml:bounds>
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										<ml:apply>
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			<math optimize="false" disable-calc="false">
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Determinate Coeff's:<sp count="3"/>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">(Eq. #40 pp17)<sp count="2"/>?????</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">(Eq. 5.53<sp count="2"/>pp163)<sp count="2"/>???</p>
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				<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Section 4.3.3<sp count="2"/>External Preassure<sp count="2"/>(pp19):</p>
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					<tab/>(Jones' Equation)<sp count="3"/>
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