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# HomeWork10 - HW#10 Chajes Problem 7-1 Using the energy...

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HW#10, Chajes, Problem 7-1 Using the energy method, investigate the behavior of the one-degree-of-freedom model of a curved plate shown in Fig. P7-1 . The model consists of four rigid bars pin connected to each other and to the supports. At the center of the model two linear rotational springs of stiffness / C M θ = connect opposite bars to each other. Also, each of the two transverse bars contains a linear extensional spring of stiffness K. Determine the load-deflection relation for the finite deflections when the load is applied (a) concentric with the axis of the longitudinal bars. (b) eccentric to the axis of the longitudinal bars. Which buckling characteristics of a curved plate do these models demonstrate? Fig. P7-1 Idealized cylindrical shell 1

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Model Analysis I Using the energy method, determine the critical load for the one-degree-of-freedom model of a curved plate (cylindrical shell) shown in Fig. 9.14 The model consists of four rigid bars pin connected to each other and to the supports. At the center of the model two linear rotational springs of stiffness C M / θ = connect opposite bars to each other. Also, each of the two transverse bars contains a linear extensional spring of stiffness K . Solution Since the coupling of in-plane and flexural actions cannot be uncoupled in a cylindrical shell, buckling of a cylindrical shell must consider this interaction. The best way to accomplish this is to develop a general large displacement formulation for the cr P and taking a limiting value as d approaches to zero by applying the L’Hspital rule. This procedure is shown at the end of Solution (a) of Model Analysis II. Model Analysis II Using the model determine the load-deflection relation for finite deflections when the load P is applied (a) concentric with the axis of the longitudinal bars, (b) eccentric to the axis of the longitudinal bars. Which buckling characteristics of a curved plate do these models demonstrate? Solution (a) ( 29 2 2 2 2 2 2 1 cos 2 2 2 1 2 1 1 L d d L L L d L L L L δ φ - = - = - - = - = - - (a) 2
The horizontal translation of each extension bar at the vertex / 2 δ will be neglected in all subsequent calculations as was done in Chapter 6. As a result of this, the angle change will be depicted somewhat greater than the actual change. The rotation of the extension bar is then 1 1 2 2 2 2 tan tan h h d L h L h θ - - - = - - - (b) The total strain energy stored in the deformed body is r e U U U = + (c) The strain energy due to rotational springs is ( 29 ( 29 ( 29 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 r U C C C C C φ θ φ θ φ θ = + = + = + (d) where 1 sin d L φ - = (e) The length of the extension bar after deformation is obtained from the Pythagoras theorem.

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HomeWork10 - HW#10 Chajes Problem 7-1 Using the energy...

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