After completing this course, you’ll have developed a complete workflow for the modelling and analysis of non-linear 3D cablenet structures. We’ll expand beyond what we covered in our study of 2D non-linear cables by adapting the tools we built previously, for the analysis of non-linear 3D cablenet structures. However, instead of just modifying our existing code, we’ll spend time fleshing out a complete workflow that takes you from initial form-finding of cablenet geometry, right through to iterative stiffness method simulation to identify the cablenet deflections and tension profile.

Introduction and course overview

Course prerequisites

Moving to JupyterLab

Cablenet structures

Section 2 overview

Building the 3D transformation matrix

Extending the non-linear stiffness matrix

Section 3 overview

Generate a simple cablenet

Extending our Blender export scripts

Visualising axial forces and deflections in Blender

Section 4 overview

Updating our notebook structure

Basic updates to our preprocessing code

Implementing a basic 3D plot

Fixing the 3D scale problem

Implement support type visualisation

Implementing 3D arrows

Implementing 3D annotations

Building the transformation matrix

Calculating the pre-tension force vector

Building the stiffness matrix

Calculating the internal force system and axial forces

Recap of the main loop logic

Updating the results plot - phase 1

Updating the results plot - phase 2

Updating the results plot - phase 3

Using Pandas DataFrames to represent our data

Visualising deflection and axial force data in Blender

Section 5 overview

Model generation in Blender

Solving the antenna tower

Saving and exporting all simulation data

Building a results viewer notebook

Section 6 overview

Simulation initial geometry in Blender - Part 1

Simulation initial geometry in Blender - Part 2

The relationship between form-finding and pre-tension

Procedural modelling with geometry nodes [Optional]

Section 7 overview

Accommodating different cable types

Generating initial geometry and pre-tension

Analysis iteration 1 - Establishing pre-tension

Analysis iteration 2 - Adding props and stay cables

Section 8 overview

Form-finding the initial geometry

Roof pre-tension simulation

Full structure pre-tension simulation

Applying load to the pre-tensioned structure

Course wrap up

Blender data-viz - Structure setup & force data

Blender data-viz - Blocking out the code

Blender data-viz - The transformation matrix

Blender data-viz - Generating materials from force data

Learn how to combine parametric modelling, exploratory form-finding and iterative analysis techniques to simulate 3D tensile structures.

Modelling and Analysis of Non-linear Cablenet Structures using Python and Blender

In this course, you’ll build an iterative numerical solver for cable and truss structures that exhibit geometric nonlinearity due to large deformations. This course focuses on building the understanding and tools to analyse structures that undergo large deflections when loaded. Large changes to the geometry of a structure can alter the internal stress distribution. This is known as geometric non-linearity and requires a more sophisticated solution strategy. We’ll place particular emphasis on cable and catenary structures as these are classic examples of structures whose deformation under load can lead to geometric non-linearity.

Deriving a linear heavy cable equation

Accounting for cable self-weight

Problem-specific boundary conditions

Solving for max cable tension

What is non-linear structural behaviour?

Large deflections and geometric non-linearity

An iterative solution strategy

The linear stiffness matrix

Additional force due to large deflections

The local non-linear stiffness matrix

The global non-linear stiffness matrix

Initial setup and data import

Plotting the initial configuration

Blocking out the main convergence loop

Building the transformation matrices

Adding pre-tension to each member

Solving for displacements

Updating the internal force system

Building a convergence test function

Calculating axial forces

Allowing for smaller external force increments

Generating a text summary output

Adding self-weight calculation

Plot setup and data selection

Plotting the undeformed structure

Building a colour scale

Plotting the deformed structure

Adding axial force labels

Plotting the applied forces

Plotting the reactions

Exploring the convergence behaviour

Modelling the cable with large axial stiffness

Introducing non-linearity by reducing the axial stiffness

Linear vs. Non-linear comparison for a simple truss

Simulating initial catenary geometry

Basic geometry data export

Exporting cable definitions

Exporting restraint data

Exporting force location data

Section 9 overview

Modifying our code for different element types

Analysing a combined cable and bar structure

Removing slack cable elements

Antenna tower - modelling and analysis

Course wrap up & Certificate of Completion

How can Blender help us?

Downloading and installing Blender

Blender overview and interface

Object versus edit mode

Rectilinear modelling

Organic modelling

Build an iterative solution toolbox to analyse structures that exhibit geometric non-linearity due to large deflections.

Non-linear Finite Element Analysis of 2D Catenary & Cable Structures using Python

Understanding non-linear structural behaviour and common geometric and material non-linearities

Non-Linear Analysis

In part 2 of this tutorial series on plastic collapse analysis, we explore plastic analysis of portal frames through worked examples.

Plastic Analysis of Frames – A Complete Guide – Part 2 | EngineeringSkills.com

Build on what we learned in part 1 to determine the critical collapse load for portal frame structures

Plastic Analysis of Frames – A Complete Guide – Part 2

This tutorial focuses on plastic analysis methods and determining the collapse load factors for determinate and indeterminate structures.

Plastic Analysis and Plastic Collapse – A Complete Guide – Part 1 | EngineeringSkills.com

Use plastic analysis to determine the collapse load factors for determinate and indeterminate structures.

Plastic Analysis and Plastic Collapse – A Complete Guide – Part 1

In this tutorial we'll explore P-Delta analysis, a geometric non-linearity that can lead to large deflections in slender structures

P-Delta Analysis and Geometric Non-linearity | EngineeringSkills.com

Free Download - The Complete Python Code for this Tutorial

In this tutorial, we'll explore P-Delta analysis, a geometric non-linearity that can lead to large deflections in slender structures

P-Delta Analysis and Geometric Non-linearity


Welcome to this tutorial series on plastic bending behaviour. This is part one of a [two part series](/posts/yielding-plastic-deformation-and-moment-redistribution-in-beams/) on plastic beam bending behaviour. In this post we’ll look at how the stress-strain curve for ductile materials in uniaxial tension maps onto the bending behaviour of beams and gives rise to plastic hinge formation.

In this post we’ll tie together some simple concepts such as yield stress and yielding, bending stress, moment capacity and plastic hinge formation. So let’s get started.

## Plastic Behaviour in Beams

So why is plastic behaviour so important to understand? It’s probably fair to say that most of our engineering analysis assumes linearly elastic behaviour. When we’re studying mechanics of materials, we almost always spend most of our time focusing on linearly elastic material behaviour. But in reality, if we limit our designs to purely elastic behaviour, we’re leaving a lot of structural capacity untapped. **Structures very often have more load carrying capacity than a linearly elastic analysis suggests**. In this post we’ll explore this reserve capacity.

<Callout type="info">
	📌 In a linearly elastic structure, stress remains linearly proportional to
	strain with **Young’s Modulus** or Modulus of Elasticity as the constant of
	proportionality.
</Callout>

In an elastic structural analysis we assume that all sections within the structure remain **linearly elastic**. In other words, at any point in the structure, **the strain remains linearly proportional to the stress at that point**. In a plastic analysis this is not necessarily the case. In a beam for example, the cross-section at one or more locations in the beam may no longer behave linearly elastically. Such cross-sections are said to be in the process of ‘**yielding**’.

**When a cross-section has fully yielded, a** plastic hinge **forms**. From the point of view of the overall stability of the structure, this is a really interesting point. If you’re not careful this can also be a very dangerous point. The formation of one plastic hinge reduces the degree of redundancy in a structure by 1. **If your structure has no redundancy, then plastic hinge formation will lead to collapse**.

Consider the following example of a simple statically indeterminate structure.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-1.png"
	alt="Propped cantilever | EngineeringSkills.com"
	width="350"
	height="186"
/>

This propped-cantilever has 4 potential reactions and is therefore statically indeterminate to one degree. As a result it could withstand the formation of one plastic hinge without collapsing. If a plastic hinge were to form at support A for example as shown below, the structure would be reduced to a statically determine simply supported beam.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-2.png"
	alt="Propped cantilever with hing | EngineeringSkills.com"
	width="252"
	height="300"
/>

So there are really two load capacities we should be aware of for a structure. The **Plastic load capacity** is the load under which sufficient hinges form to cause a structure to collapse. The **Elastic load capacity** is the load under which any section in the structure exceeds its elastic limit.

<Callout type="warning">
	⚠️ Only when sufficient plastic hinges have formed to exhaust all redundancy in
	a structure, will it collapse.
</Callout>

## Stress-strain Curve and Plastic Deformation

So far we’ve loosely mentioned this concept of hinge formation and the resulting loss of a degree of redundancy. To better understand this, we need to take a closer look at exactly what is happening when a hinge forms in a beam.

**For a plastic hinge to form, we must be dealing with an elastoplastic material.** Or in other words, none of what we’re discussing in relation to plastic hinges applies to brittle materials. So, structural steel (ductile) definitely can develop plastic hinges. Glass (brittle) cannot, mass concrete (brittle without reinforcing steel) cannot but reinforced concrete can under the right circumstances because of the presence of reinforcing steel.

So, for the purposes of our discussion let’s assume we’re dealing with structural steel. Steel is an elastoplatsic material that follows Hooke’s law until the yield stress, $\sigma_y$

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-3.png"
	alt="stress-strain-curve-moment-redistribution | EngineeringSkills.com"
	width="400"
	height="324"
/>

-> Plastic yielding is characterised by an increase in strain at a fixed magnitude of stress.

Structural steel is a great example of an elastoplastic material:

- It has a well defined yield point
- It undergoes large strains during yielding
- It exhibits the same material behaviour in tension and compression

Consider the stress-strain curve for structural steel (in uniaxial tension) in a bit more detail below:

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-4.png"
	alt="Stress-strain-curve | EngineeringSkills.com"
	width="800"
	height="502"
/>

The linearly elastic range is exceeded when the tensile stress exceeds the proportional limit. This is very quickly followed by the onset of the plastic or yielding region when the tensile stress reaches the yield stress, $\sigma_y$. This plastic region is central to the formation of plastic hinges.

Of less interest but still worth noting are the regions following the plastic region.  The strain hardening zone terminates at the ultimate stress, $\sigma_u$ and is immediately followed by the necking region prior to rupture of the material.

The key point to take from this graph is that the plastic region is characterised by an increase in strain at a constant value of stress (the yield stress). This is the definition of plastic behaviour!

Our task now is to take this material behaviour, and map it onto beam bending behaviour. In other words, what how does this plasticity manifest in a beam undergoing bending?

## Stress-strain Behaviour & Plastic Hinge Formation

Consider a beam under the influence of a progressively increasing external point load (and resulting internal bending moment) at the mid-span point. We’ll refer to the internal moment of resistance, induced by the point load as an ‘applied’ moment $M_a$.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-5.png"
	alt="Simply-supported beam with point load | EngineeringSkills.com"
	width="350"
	height="270"
/>

Now consider the stress in the beam cross-section at $x=L/2$ and let $\sigma_y$ denote the yield stress for the beam material. From the standard bending stress equation we know the moment at which yielding of the beam material will occur,

$$
M_y = \frac{\sigma_y\:I}{c}
\tag{1}
$$

where $I$ represents the second moment of area for the beam cross-section and $c$ represents the distance from the extreme beam fibre to the neutral axis (N.A)

💡 Remember the neutral axis is the point in the cross-section at which the longitudinal/normal stress due to bending is zero

We can now consider the different stages of behaviour the cross-section goes through as the point load is increased and we observe elastoplastic behaviour. In particular we’ll focus on the stress and strain distribution at the cross-section located at $x=L/2$, the point of maximum moment.

### Stage 1: Applied moment $M_a$ < Yielding moment $M_y$

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-6.png"
	alt="Stage 1 M_a<M_yield | EngineeringSkills.com"
	width="700"
	height="359"
/>

At this stage, the cross-section behaves elastically and the stress, $\sigma$ remains below the yield stress, $\sigma_y$. This is the simplest form of cross-section behaviour to model and is what you’ve probably spent most of your time considering. **As we’ll see, this linear elastic model of cross-section behaviour is somewhat inefficient as it leaves the material strength within the plastic range of behaviour untapped.**

### Stage 2: Applied moment $M_a$ = Yielding moment $M_y$

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-7.png"
	alt="Stage 1 M_a<M_yield | EngineeringSkills.com"
	width="700"
	height="365"
/>

When the applied bending moment (resulting from increasing external load P remember) is increased such that the maximum normal stress due to bending $\sigma$, is equal to the material yield stress $\sigma_y$, the cross-section has reached the limit of the elastic range. Any further loading will cause partial yielding of the cross-section.

### Stage 3: Applied moment $M_a$ > Yielding moment $M_y$

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-8.png"
	alt="Stage 1 M_a<M_yield | EngineeringSkills.com"
	width="700"
	height="365"
/>

If the moment is increased further, more fibres reach the yield stress $\sigma_y$, **but do not exceed it**. However, strains will continue to increase beyond the yield strain. Therefore the relationship between stress and strain is no longer linear (i.e. stress is no-longer simply a constant multiple of strain). Remember, this is the definition of plastic behaviour – increasing strain for a constant value of stress.

The longitudinal stress is limited and cannot increase beyond the yield stress. So as more load is applied, more and more fibres reach this limit, which has the effect of ‘plastic regions’ developing at the outer edges of the cross-section and progressively growing towards the centre.

We end up with an elastic core, (shown in blue in the image below, with plastic regions at the outer edges of the cross-section – shown in red). At the same time the strain (and therefore rotation) at this cross-section continues to increase (green below).

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-10.png"
	alt="Elastic core with plastic outer regions | EngineeringSkills.com"
	width="700"
	height="431"
/>

The strain must continue to increase as long as there is increasing deflection of the beam and therefore rotation at the cross-section. You might recall from basic beam bending theory that the relationship between strain at a section and curvature is,

$$
\epsilon = -\kappa y = \frac{y}{\rho}
\tag{2}
$$

where, $\kappa$ is the curvature of the beam, $\rho$ is the radius of curvature and $y$ is the distance from the neutral axis to the point at which the strain occurs.

This relationship relates the strain behaviour at a cross-section to the deflection (specifically rotation) of the beam and explains why the strain must continuing to increase despite the fact that the stress remains constant.

### Stage 4: Applied moment $M_a$ = Yielding moment $M_y$

If the applied force is increased still further, the elastic core will continue to reduce in size until it disappears completely. **At this point the section has fully yielded and can withstand no further increase in bending moment**.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-9.png"
	alt="Stage 1 M_a=M_plastic | EngineeringSkills.com"
	width="700"
	height="324"
/>

💡 The bending moment at which this point is reached (elastic core disappears) is called the Plastic Moment. This represents the full moment capacity of a cross-section.

If the bending moment is increased beyond the plastic moment, the section behaves like a (plastic) hinge, allowing rotation to occur. We can see from the image below, that if the maximum moment in the statically determinate simply supported beam reaches the plastic moment, hinge formation would lead to the structure becoming a mechanism and collapsing.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-12.png"
	alt="SS beam hinge formation and collapse | EngineeringSkills.com"
	width="700"
	height="278"
/>

We can now summarise the progression of the cross-section through various states from, from fully elastic to fully plastic,

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-11.png"
	alt="SS beam hinge formation and collapse | EngineeringSkills.com"
	width="800"
	height="556"
/>

From this discussion we can see that if we have a ductile material, designating the yield moment, (the moment at which fibres in the section first reach their yield stress), as the maximum allowable moment is somewhat conservative. In reality, the cross-section has a further reserve of capacity up until full yielding of the complete section occurs.

The formation of outer plastic regions while an inner elastic core persists is known as elastoplastic bending behaviour. When we have exhausted the elastoplastic capacity of the material and the section becomes fully plastic, a plastic hinge is said to have formed. Any further loading of the structure would cause free rotation at the location of the plastic hinge.

Now that we have a good grasp conceptually of what’s going on when a hinge forms, [in the next post](/posts/yielding-plastic-deformation-and-moment-redistribution-in-beams/), we’ll move on to determine the plastic moment capacity of a cross-section. We’ll also think a little more about moment redistribution in statically indeterminate structures.

---


In this tutorial on plastic bending behaviour we'll look at how the stress-strain curve for ductile materials gives rise to moment redistribution in beams

The Stress-Strain Curve & Plastic Hinges in Beams (1/2) | EngineeringSkills.com

In this tutorial, we'll look at how the stress-strain curve for ductile materials gives rise to moment redistribution in beams

The Stress-Strain Curve & Plastic Hinges in Beams (1/2)


In the [previous post in this series](/posts/stress-strain-curve-moment-redistribution/) we introduced the idea of elastoplastic bending. We showed how the normal/bending stress distribution transitioned from linearly elastic to fully plastic. We saw that a fully plastic cross-section behaved just like a hinge allowing free rotation if further loading was applied to the structure. We finished that post with the clear understanding that a cross-section has additional plastic moment capacity (ductile materials only!)

In this tutorial, we’re going to work out exactly how to determine the plastic moment capacity of a cross section. We’ll also explore the concept of moment redistribution with an illustrative example. By the end of this post you’ll be able to calculate the plastic moment capacity of any cross-section and understand in detail how moment redistribution occurs in a structure.

## Plastic Moment Capacity and the Shape Factor

Once we’ve established the fact that a cross-section can go beyond its elastic limit and sustain further loading after it has begun to yield, the obvious question is how much more load can it withstand? We can start by considering a cross-section which has reached its plastic limit.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-13.png"
	alt="Plastic moment capacity | EngineeringSkills.com"
	width="700"
	height="351"
/>

We know that to satisfy equilibrium, the total force acting on the cross-section must be zero, therefore,

$$
F_t = F_c
\tag{1}
$$

We know force is simply the product of stress and area (over which the stress acts), so,

$$
\sigma_yA_1 = \sigma_yA_2
\tag{2}
$$

From this we can conclude that when a cross-section has fully yielded, the neutral axis divides the cross-section into two equal areas. So, if the z-axis is not an axis of symmetry (and it’s not in the case of the trapezoidal cross-section shown above), the neutral axis under plastic conditions will be in a different location to the neutral axis under elastic conditions. We can now state an expression for the plastic moment as,

$$
M_p=F_c \bar{y}_1 + F_t \bar{y}_2
\tag{3}
$$

Now since,

$$
F_t = F_c = \frac{\sigma_y A}{2}
\tag{4}
$$

We can state that **the plastic moment is**,

$$
M_p = \frac{\sigma_y A}{2}(\bar{y}_1 + \bar{y}_2)
\tag{5}
$$

We now define the plastic section modulus as,

$$
S = \frac{A}{2}(\bar{y}_1 + \bar{y}_2)
\tag{6}
$$

And we can state the plastic moment as,

$$
M_p = \sigma_yS
\tag{7}
$$

It should be noted that the plastic modulus is purely a function of the cross-section shape.

### Shape Factor

We know from basic beam bending theory that the elastic moment is given by,

$$
M_y =\frac{\sigma_y I}{y}
\tag{8}
$$

and that the elastic modulus is given by,

$$
Z=\frac{I}{y}
\tag{9}
$$

Now we define the ratio of plastic moment, $M_p$ to yield moment $M_y$  as the shape factor, $f$ and note that this ratio is ultimately a function of the cross-section shape,

$$
f=\frac{M_p}{M_y} = \frac{S}{Z}
\tag{10}
$$

<Callout type="info">
	💡 The shape factor is a measure of how much **reserve strength** is in the
	cross-section after yielding begins.
</Callout>

The shape factor is a helpful indicator of how much more loading can be sustained after yielding but before a plastic hinge is formed. Remember, it is simply a function of the shape of the cross-section and is independent of the material properties of the cross-section.

## Calculating the Plastic Moment Capacity

Now we’ve covered all of the theory, let’s actually calculate the plastic moment capacity and shape factor of a cross-section. To make things a little more interesting we’ll consider a compound cross-section consisting of an I-beam with a steel plate welded to the bottom flange. We’ll assume a steel yield stress of $\sigma_y = 350 \:N/mm^2$.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-14.png"
	alt="Plastic moment capacity example | EngineeringSkills.com"
	width="350"
	height="396"
/>

You can watch me walk through the full solution to this question in the video below (the thought of typesetting the full thing in this post was too much to take!) But the outline steps are as follows:

- We need to calculate the second moment of area using the parallel axis theorem. But as this is a compound section and not symmetrical about the neutral axis, we’ll need to first locate the neutral axis, by taking area moments about an arbitrary axis.
- Armed with the second moment of area and neutral axis location, we can plug these into the engineer’s bending equation (equation 8 above) to determine the moment required to cause yielding, $M_y$ .
- Then we can move on and determine the location of the equal area axis. Remember the neutral axis becomes an equal area axis (and may shift location) in the fully plastic condition.
- Once we have the equal area axis, we can work out the plastic moment capacity and apply equation (10) to determine the shape factor.

It’s worth you trying to solve this yourself first before watching me solve it in the video below.

<Youtube id="J7hcrrrHoLY" />

## Plastic Hinges in Beams and Moment Redistribution

Now we’re going to expand our study of elastoplastic behaviour beyond the cross-section to look at hinge formation in a beam. Consider the simply supported beam below subject to an increasing point load at mid-span. By examining the bending moment diagram, we can map out and visualise the formation of a plastic hinge under the point load.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-15.png"
	alt="Hinge formation in a beam | EngineeringSkills.com"
	width="700"
	height="897"
/>

We can clearly see how the bending moment diagram can be used to identify the extent of the elastoplastc region in the beam. As we now know, once the entire section becomes plastic (as indicated at the mid-span in the beam above) a plastic hinge forms.

Now we’ll consider how hinge formation allows moment to be redistributed in a statically indeterminate structure. We demonstrate the fundamentals of this behaviour by considering a simple (indeterminate) propped cantilever.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-16.png"
	alt="Propped cantilever plastic hinge 1 | EngineeringSkills.com"
	width="350"
	height="214"
/>

It can be shown that the bending moments at A and C are:

$$
M_A = \frac{3PL}{16}
\tag{11}
$$

$$
M_C =\frac{5PL}{32}
\tag{12}
$$

We’re going to set ourselves a simple task; to determine the value of the applied load, $P$ at which the structure collapses. In doing so we’ll need to take account of plastic hinge formation and moment redistribution.

We can start by stating some basic assumptions:

- Let the span $L=2 \:m$
- Let the yield moment $M_y = 12\: kNm$
- Let the plastic moment $M_p = 15 \:kNm$

Let’s also assume the the beam must accommodate an applied **working load** $P = 25 \:kN$.

### Analysis for P=25 kN

Under normal working conditions, the moments at A and C are:

$$
M_A = \frac{3\times 25\times 2}{16}=9.4 \:\:kNm
\tag{13}
$$

$$
M_C = \frac{5\times 25 \times 2}{32} = 7.8 \:\:kNm
\tag{14}
$$

Both moments are less than the yield moment therefore the beam is safe under working conditions. So let’s push things further – remember we want to know the load at which the beam collapses.

### Analysis for P=32 kN

Again determining the moments in the structure:

$$
M_A = 12 \:\:kNm = M_y
\tag{15}
$$

$$
M_C = 10\:\:kNm
\tag{16}
$$

Therefore the cross-section at location A has reached its yield limit; i.e. the extreme fibres in the cross-section at this location have reached their yield stress. In a linearly elastic analysis we would call this the maximum allowable load and end things here. But as we know, this beam can take more, so let’s push things further still.

### Analysis for P=40 kN

In this case, the moments are:

$$
M_A = 15 \:\:kNm = M_P
\tag{17}
$$

At this point, a plastic hinge has formed at A. Any further loading will cause free rotation to occur at this location.

$$
M_C = 12.5\:\:kNm >M_y
\tag{18}
$$

At C, the section has exceeded its elastic limit and yielding has started. At location C the section is undergoing elastoplastic bending.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-17.png"
	alt="Propped cantilever plastic hinge 2 | EngineeringSkills.com"
	width="400"
	height="224"
/>

It’s important to remember that the structure started out as indeterminate to 1 degree i.e. it had one redundancy. It has just lost this redundancy with the formation of a plastic hinge at A. The structure will not collapse yet, but if one more hinge forms, the structure will become a mechanism and will collapse.

### Analysis for P=45 kN (Moment Redistribution!)

At this load, the moment at A remains 15 kNm. Remember, the section can’t withstand any more moment once a hinge has formed. Any further loading after hinge formation simply causes rotation about the plastic hinge.

**So if any additional load is applied, we must have moment redistribution in the beam**. This is inevitable; the load is continuing to increase, but a hinge has formed at A allowing the section to rotate at that location. If the structure is to avoid collapse, the increased moment within the structure must be redistributed from the support at A and into the span, further increasing the moment at C.

The formation of a plastic hinge at A means that the structure now behaves as a simply supported beam. In other words, all of the load applied between P=40 kN and P=45 kN is resisted by the structure as if it were a simply supported beam.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-18.png"
	alt="Propped cantilever plastic hinge 3 | EngineeringSkills.com"
	width="350"
	height="176"
/>

Now determining the moment at C:

$$
M_c = 12.5 + 2.5 = 15 \:\:kNm = M_P
\tag{19}
$$

We can see that at this value of applied load, a hinge will form at C. The structure now becomes a mechanism and collapses.

<Image
	src="/images/posts/stress-strain-curve-moment-redistribution/image-19.png"
	alt="Propped cantilever plastic hinge 4 | EngineeringSkills.com"
	width="400"
	height="317"
/>

The load at which collapse occurs is 45 kN while the load at which yielding first occurs is 32 kN. So this structure could withstand approximately 40% more load before collapse than an elastic analysis would suggest.

This ability to safely sustain loading after plastic hinge formation is often utilised in structural design. Large span steel portal frames are often designed to develop plastic hinges. Care must obviously be taken to ensure sufficient redundancy remains to ensure collapse does not occur.

## Plastic Collapse Analysis

We can define a load factor $\lambda_i$ as the collapse load for mechanism $i$, divided by the structures’ working load,

$$
\lambda_i = \frac{collapse\:\:load \:\:for \:\:mechanism \:\:i}{working \:\:load}
\tag{20}
$$

For a structure with multiple collapse mechanisms, the collapse load factor is the smallest value of the load factor – the load factor at which the structure will collapse. For our example above, the collapse load factor is,

$$
\lambda_C = \frac{45}{25} = 1.8
\tag{21}
$$

We might explore plastic collapse analysis in another tutorial series. But for now you should have a good understanding of the material and cross-section behaviour that leads to collapse of ductile structures and the additional capacity that exists, after material yielding but before plastic hinge formation.

In case you missed it, the first tutorial in this 2 part series [can be found here](/posts/stress-strain-curve-moment-redistribution/).


By the end of this tutorial you’ll be able to calculate plastic moment capacities and understand in detail how moment redistribution occurs in a structure.

Yielding, Plastic Deformation and Moment Redistribution in Beams (2/2) | EngineeringSkills.com

Learn how calculate plastic moment capacities and how moment redistribution occurs in a structure

Yielding, Plastic Deformation and Moment Redistribution in Beams (2/2)

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