Week 9 Challenge- Non-Linear analysis and Critical Load

1) Explain the following levels of analysis

Ans :-

a) First Order Elastic analysis - In first-order analysis the stiffness of the structure is assumed to be constant and unaffected by changes in the geometry of the structure when it is loaded. This is the standard assumption of linear elastic first order analysis.

b) Second order Elastic analysis - Second order analysis takes into account how the structure deforms while loads are being applied on it. Numerically speaking, all load is divided into smaller portions and for each load portion, the stiffness of the structure changes.

c) First order inelastic analysis - In first-order inelastic analysis the equations of equilibrium are written in terms of the geometry of the undeformed structure. Inelastic regions can develop gradually or, if the plastic hinge concept is adopted, as abrupt changes in the structure's response.

d) Second order inelastic analysis - Two simplified methods for second‐order inelastic analysis of steel frames, termed the modified plastic‐hinge and the beam‐column strength approaches, are presented. These analysis models are comparable to elastic‐plastic‐hinge analysis in efficiency, and yet, they alleviate the problems associated with overprediction of stiffness and strength by the usual elastic‐plastic‐hinge analysis methods. The modified plastic‐hinge method is based on simple modifications to the elastic‐plastic‐hinge model, which account for the degradation in stiffness as the cross‐section strength is approached at critical locations along the member length. The beam‐column strength method is similar to the modified plastic‐hinge approach, except that in this model, equations for the strength of the overall member are employed rather than expressions for the cross‐section strength. The performance of these analysis models is contrasted with results from elastie‐plastic‐hinge and refined plastic‐zone analysis solutions as well as results from experimental tests. Recommendations are provided with regard to the acceptability of simplified analysis models for use in predicting inelastic load‐deflection behavior and maximum strength.

2) Give examples for the following sources of Non Linearity

a) Geometry - Geometrical nonlinearity associated with the change in geometry of the structure under load.

Example - Elastic buckling of column, fishing rod.

b) Material - Material nonlinearity related to the onset and spread of plasticity.

Example - Large strain (visco) elasto-plasticity and hyperelasticity (rubber and plastic materials).

c) Contact - Contact non linearity occurs when, due to the deformation of one or more parts in contact.

Example - When say a round object falls on an elastic deformable surface, gear teeth, a cylindrical object rolls over a surface.

3) Evaluate the critical load for the column shown below following small displacement theory

Ans :-

In order to analyse this structure, we’re going to use the differential equation of the defection curve in which is the internal bending moment at a given cross-section, represents the lateral deflection of the column and the distance along the length of the column. is referred to as the flexural rigidity and is the product of Young’s modulus and the second moment of area of the cross-section .

First consider the structure, its deflected shape and the free-body diagram that results from cutting the structure at a distance from the bottom support…

Taking moments about point A yields:

Substituting back into the differential equation of the deflection curve and rearranging slightly yields:

This is a linear, homogeneous, 2nd order differential equation with constant coefficients. It also happens to be an equation that models the deflected shape of our structure and contains , the axial load. Following the same approach as for the idealised structure discussed previously, if we solve this equation we can determine , the value of applied load at which it is balanced by the column’s flexural bending resistance.

So we’re faced with trying to solve a differential equation. At this point, it’s helpful to be able to point to the general solution of this form of differential equation:

Boundary condition 1 :-

At (base of the column at pin support), (the lateral deflection must equal zero. Imposing these conditions on our general solution yields:

Therefore

As a result our solution reduces to:

Boundary condition 2 :-

At (top of the column at pin support), (lateral deflection also zero). Again, imposing these boundary conditions yields,

So, either equals zero or equals zero. If equals zero, the equation is satisfied for any value of . This doesn’t really move us forward…we can think of this solution as the trivial solution. Of more interest is the case where:

Knowing what we know about functions, this equation can only be true, when,

We can dismiss the case where,

In this scenario, would need to equal zero (as all other terms have non-zero values by definition) and we know this is not the case. So for a pinned-pinned column, our solution is:

where … Finally, we can rearrange this equation to make the subject of the equation:

This equation represents an infinite series of buckling loads. The lowest one is the critical buckling load, also known as the Euler Buckling Load .

4) Explain the concept of equilibrium with examples

a) Stable Equillibrium - equilibrium is said to be stable if small, externally induced displacements from that state produce forces that tend to oppose the displacement and return the body or particle to the equilibrium state.

Example - Weight suspended by a spring, brick lying on a level surface.

b) Neutral Equiilibruim - A system is in neutral equilibrium if its equilibrium is independent of displacements from its original position.

Example - Marble on a flat horizontal surface.

c) Unstable Equillibruim - An object is in unstable equilibrium if, when a disturbing force is applied, the centre of gravity is lowered and the object moves away from its original position.

Example - A marble balanced on the top of an inverted bowl.

5) Develop a matlab function to implement evaluation of critical load capacity of columns for different boundary conditions ( eg: Pin-Pin , Fixed-Fixed , Fixed- Free )

Ans :-

Programme is as below -

function [Global_Stiffness_Matrix]=Computeglobalstiffnessmatrix()

[node_coord,element_no,element_node,length_element,element_geometry,element_dof]=Getelementinfo();

boundary_conditions=Getboundaryconditions();

node_loads=Getnodalloads();
no_elements=size(element_no,1);
no_nodes=size(node_coord,1);

Global_Stiffness_Matrix=zeros(3*no_nodes,3*no_nodes);
displacement_matrix=zeros(3*no_nodes,1);
force_matrix=zeros(3*no_nodes,1);

%boundary condition
all_dof=1:3*no_nodes;
dof_specified=[];
for i=1:no_nodes
    n_dof=3*(boundary_conditions(i,1)-1)+boundary_conditions(i,2)
    dof_specified=[dof_specified n_dof];
    displacement_matrix(n_dof)=boundary_conditions(i,3);
end
free_dof=all_dof;
free_dof(dof_specified)=[];

%compute nodal loads
for j=1:size(node_loads,1)
    force_matrix(3*(node_loads(j,1)-1)+node_loads(j,2))=node_loads(j,3);
end

%gloabal stiffness matrix
for k=1:no_elements
    ele_nodes=elements_node(k,:);
    node_xy=node_coord(ele_nodes,:);
    A=element_geometry(k,1)*element_geometry(k,2);
    I=element_geometry(k,1)*element_geometry(k,2).^3/12;
    length=length_element(k);
    k_element=Frame_Stiffness_Matrix(node_xy,A,I,length_element);
    eldofs=element_dof(k,:);
    Global_Stiffness_Matrix(eldofs,eldofs)=Global_Stiffness_Matrix(eldofs,eldofs)+k_element;
end

displacement_matrix(free_dof)=Global_Stiffness_Matrix(free_dof,free_dof)\(force_matrix(free_dof)-Global_Stiffness_Matrix(free_dof,dof_specified)*displacement_matrix(dof_specified));
force_matrix(dof_specified)=Global_Stiffness_Matrix(dof_specified,:)*displacement_matrix;
disp('Displacement Vector');
disp(displacement_matrix);
disp('Force Vector');
disp(force_matrix);
end

Explanations of codes -

function - to create function with given inputs.

Getelementinfo - to take element data from the function.

Getboundaryconditions - to take boundary conditions from the fucntion.

Gernodalloads - to take nodal loads from the given function.

size - to identify number of elements and nodes

zeros - to create zeros matirx .

for - to create loop data with respect to given input.

end - to end the loop.

disp - to display the results.

end - to end the function.

6) Develop an algorithm to implement matrix approach in second order elastic analysis.

Ans :-

As inputs are nodes, element, moment of inertia, section area, section length.

[Ke]-[Kg]=[Kt] ...K(stiffness matrix) is not constant.

[Ke]+[Km]=[Kt] ...change with every iteration

a) breaking load increment as second increment of force vector

Ki[d(delta)i]=[dPi]=[d(lamda)p]

b) for i=1 to n elements

[Ke+Kg][d(delta)i]=[dPi]

c) [stiffnesss matrix]=d(delta)i calculation of dPi