CE 572 Stability of Structures Home Work #1 1. Consider the system shown below with two equal rigid bars with a hinge between them and rotational and transverse springs at the hinge point. Considering the deformation to be large, set the Total Potential Energy of the system in the indicated coordinate (θ). a. Determine the bifurcation buckling load, Pcr. b. Determine the expression for post buckling path (i.e. for θ ≠ 0). Comment on the state of equilibrium for the secondary path at bifurcation load, if (i) k = 0, (ii) β = 0 and (iii) kL2 = 8β. Comment on state of equilibrium on the secondary path at θ = π/4, θ = π/2 and θ = 2π/3 in terms of type of buckling c. What conditions do we need to apply to ensure stability at the critical load. 2. Solve the stability conditions of the column system shown at side using large deflection analysis: by (a) classical approach using free body diagrams, and (b) energy approach. Draw the load versus deflection diagram and indicate the states for both primary and secondary equilibrium paths. If the axial load, P is applied with eccentricity, e, draw the load versus deflection diagram for the given eccentricity, e. Find the limit point buckling load for this case. Plot Pcr versus e/l. 3. Determine the equilibrium path(s) of the structure shown under the effect of axial load P. A frictionless hinge at C connects the two rigid bars AC and BC. A Linearly elastic axial spring of stiffness (k) also supports the structure at B. The structure is in a state of stree-free condition when the angle α as shown in the figure. Investigate the stability of the equilibrium path.

4. Solve the stability conditions of the column systems shown in the figures below:

5. For the 2DOF (ψ,θ) bar-spring model shown, a) Assume no imperfections (ψ0=0, θ0=0 ) and that the supporting springs are elastic. Find Pcrx (ψ mode) and Pcrθ, (θ mode) assuming small deflections. b) Determine large deflection equilibrium or energy solutions for the two modes and plot them individually on P versus ψ and P versus θ plots. Assess the stability of at least one of your individual equilibrium solutions and use your intuition to estimate stability of generated solutions. c) Transform your solutions from (2) to the P versus Δ space and plot them together. What happens when Pcrx > Pcrθ and when Pcrx < Pcrθ? d) Determine a large deflection equilibrium solution for the case of small imperfections ψ0, and θ0 and plot an example of your solution. e) Consider the possibility that the springs are elasticplastic. Plot the load at which the springs will yield in your P versus ψ, P versus θ, and P versus Δ plots. What happens when solutions cross? 6. Determine the elastic buckling loads for the following system comprising of rigid bars and hinges. Formulate the problem in terms of deflections at b and c, yb and yc. Indicate the buckled mode shapes

P

a

2P b

K

c

d 2 K

7. In the coplanar system shown, the initially vertical rod is rigid. The block to which the spring is attached slides in the incline guide and is controlled so that the spring is always horizontal. All parts have negligible mass except weight W. a. Show that the tilted equilibrium positions are characterized by

b. Sketch the curve for the two cases tanβ small (e.g., 1/20 ) and tanβ large (e.g., 10). What conclusions can you draw as to the stability of the tilted position? Give reasons. c. Show that the vertical position is stable with respect to sufficiently small disturbances so long as . 8. An ideal column is fixed at an end and fixed to a rigid bar of length a EI, L at the other end. The second end of the rigid bar is pinned on roller. Find the critical load condition and discuss the extreme cases. ( a → 0 and a → ∞).

Rigid a

P

9. Write the second order differential equations for buckling and solve for exact characteristic equation, which defines the critical load. You do not need to find the critical load.

10. For a column of length L, and constant stiffness EI which has a pin and rotational spring kφ at one end and pinned at the far end a. find the buckling load,. b. find an appropriate non-dimensional ratio between the spring stiffness and column stiffness and plot what happens in the solution as this ratio varies. The effective column length KL is the length that you would plug into the classical Euler buckling formula such that you get the correct buckling load. c. prove that KL is the distance between inflection points. 11. Consider a column which has moment of inertia αI for the lower L/2 and I for the upper L/2 with α≥1. a. Determine the buckling load under pin-pin boundary conditions, and b. demonstrate the influence of α on the solution c. buckling load for fixed (at the αI end) -free boundary conditions, and d. demonstrate the influence of α on the solution