## Plastic section modulus of a composite section

## Plastic section modulus of a composite section

(OP)

I am trying to verify cross-sectional properties calculated by Section designer in sap2000 with different base material and material of shape.

For this exercise I've defined A36 as base material and C30 as material of the shape inside the section designer.

I've verified cross-sectional properties such as A,I,S & r, except plastic section modulus Z.

I'm calculating Z for a rectangular shape as follows.

Z = (Ec/Es)b(d^2)/4

Ec/Es for my case will be equals to 1/8.

Software result for Z only, is being calculated for Ec/Es = 1/9.

Sap2000 is calculating Z like this for other cross-sections too. Does anyone have any idea why sap is doing this?

P.S. I've verified the cross-sectional properties for same base material and material of shape. My manual results does match with software results.

For this exercise I've defined A36 as base material and C30 as material of the shape inside the section designer.

I've verified cross-sectional properties such as A,I,S & r, except plastic section modulus Z.

I'm calculating Z for a rectangular shape as follows.

Z = (Ec/Es)b(d^2)/4

Ec/Es for my case will be equals to 1/8.

Software result for Z only, is being calculated for Ec/Es = 1/9.

Sap2000 is calculating Z like this for other cross-sections too. Does anyone have any idea why sap is doing this?

P.S. I've verified the cross-sectional properties for same base material and material of shape. My manual results does match with software results.

## RE: Plastic section modulus of a composite section

## RE: Plastic section modulus of a composite section

I hope the following pictures explains my problem better.

## RE: Plastic section modulus of a composite section

My Personal Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

Open Source Structural GitHub Group:

https://github.com/open-struct-engineer

## RE: Plastic section modulus of a composite section

Csi Softwares usually calculate cross-sectional properties for uncracked sections. We manipulate those results to get cracked section properties by assigning property modifiers.

Also, if sap2000 was considering cracked concrete to find Z here, then NA will have to move up quite a lot to make the tension force = compression force. Z calculated, considering concrete in compression only, will be a lot more smaller than the sap2000 value.

Fy/0.85f'c will be equals to 17.5. This will also give a very smaller value.

I'm aware that it's a composite section and flexure capacity need to be calculated accordingly. But, I can't seem to figure out how sap2000 is calculating the Z value and essentially the flexural capacity of the section.

What will be the use of section designer if sap2000 may not calculate the properties of a composite section accurately.

## RE: Plastic section modulus of a composite section

The plastic modulus is a property related to the strength weighted area above and below the plastic neutral axis, or in other words proportional to the force carried by the section above and below the plastic neutral axis. If you have different materials it's calculation is therefore related to strength not the modulus of elasticity. You'd use E values for working out the transformed stiffness.

Now you'll need to follow whatever code you're using for the ratio of strength reduction factors applied to the steel and concrete for composite design to work out the force. Ignoring the portion of the concrete in tension as Celt83 noted.

But in a way the calculations for the plastic modulus are somewhat redundant as you likely also have an axial load and you need to factor that in as well. I'm only familiar with Eurocodes in relation to this, but you actually look at the section more like you would for a reinforced concrete column. Calculating an interaction diagram accounting for the initial imperfections. Unsure how other standards treat that aspect, but I doubt it's a normal 'steel' interaction check using the equivalent of a weighted plastic section modulus times a strength of the base material. Therefore I'm doubtful using the Sap2000 result is of any value.

## RE: Plastic section modulus of a composite section

I'm coming to realize that you're right about this as well.

The goal here was to try to figure out how sap2000 was calculating the Z of a composite section, designed as a steel coulmn. My first thoughts were that, it's determining the Z by converting the composite section into a homogeneous material.

I'm using AISC code, which is mostly silent about the flexural strength of concrete filled HSS. As far as I know, it considers the interaction of axial and flexural load by specifying that the dcr should be less than 1.0.

However, I've seen the provsions of eurocode for determining the flexural capacity of concrete filled tube, which is pretty basic mechanics, and I know that, it considers the interaction of axial and flexure force using interaction diagram.

## RE: Plastic section modulus of a composite section

-Plastic stress distribution, strain compatibility, elastic stress distribution method, effective stress-strain method.

On the design/capacity side, for flexure:

It's a function of member slenderness, equations are based on plastic stress distribution for compact sections and strain compatibility for noncompact and slender ones and it's assumed:

-For compact sections, steel plastifies, concrete reaches 0.85f'c (or 0.95f'c in round sections due to better confinement), tension and compression forces are calculated to find moment capacity.

-For noncompact sections the steel is assumed to reach yield, concrete can only reach 0.7f'c

-For slender sections, the steel only reaches critical buckling stress, F_cr, concrete limited to 0.7f'c

On the design/capacity side for axial:

It's a function of the section, concrete, rebar, and slenderness. There are equations for compact, noncompact, and slender strengths, which are reduced for the global effects of slenderness.

For interaction:

-Plastic distribution for compact and strain-compatibility for noncompact/slender sections.

-4 approaches are available in AISC: AISC H1, Linear interaction, AISC Design Guide 6, Direct Interaction Method

AASHTO also has provisions regarding concrete-filled tubes and concrete encased shapes.

## RE: Plastic section modulus of a composite section

Most of what you've stated above, I found in AISC chapter I and commentary section, but some of your statements were new to me. Can you help me identified the reference of your following statements.

1. effective stress-strain method.

2. Direct Interaction Method (I'm afraid I'm hearing of this method for the first time from you.)

3. For noncompact sections the steel is assumed to reach yield, concrete can only reach 0.7f'c.

4. For slender sections, the steel only reaches critical buckling stress, F_cr, concrete limited to 0.7f'c.

## RE: Plastic section modulus of a composite section

https://www.youtube.com/watch?v=1fzqBwZKLJQ&fe... (the information I provided largely came from this presentation at NASCC last year). The effective stress-strain method is discussed at around 20:00. Direct interaction method around 46:00.

https://steeltubeinstitute.org/hss/2016/11/01/comp...