Shear stress

Introduction

In this article shear stress will be considered. The basic and the most known equation for shear stress is equation below:

średnie naprężenie styczne

Sometimes it is forgotten that stress calculated in this way are mean stress. In many cases such approach is sufficient. To be more conscious of simplification above let's try to calculate max value of the shear stress according to Żurawski equation for defined profiles.

 

1. Shear stress in flat bar

Let's start from the rectangular section. Mean stress for section of dimensions 10x100mm and shear force 10 000 N is equal 10MPa. In order to determine max shear stress value we use Żurawski equation:

 

where: T - shear force, S - first moment of area about the centroidal axis of that portion of the cross - section between the point at which shear is required and the boundary of the cross - section, Ix - second moment of area of the whole cross - section about x axis.

 

To calculate shear stress value at "y" height first moment of area is determined. It concerns area which lies above mentioned "y" coordinate, i.e.

 

 

First moment of area

 

where c - "y" coordinate of the cross - section center

 

Shear stress for rectangular section calculated according to Żurawski eqution changes from 0 to 15 MPa. It can be noticed that max value is 1.5 times bigger than

mean value.

 

Summarizing maksymalne naprężenie styczne

 

Mean and max shear stress acc. to Żurawski equation are presented below.

 

 Maximum shear stress

 

2. Shear in I section

In case of I section shear stress distribution according to Żurawski equation is presented below.

I section shear

It can be noticed that shear stress for flanges are quite small. That is why it is assumed that shear is carried primary by the web.

For example let's determine shear stress for IPE 160. Shear force 10 000 [N] is applied. Dimensions for IPE 160 are presented below.

 

 IPE 160

 

Flange first moment of area around "y" axis is equal:

 

 

 

Shear stress at point A is equal:

Shear stresses at point B are equals:

 

 

 First moment of area around "y" axis for beam area below point C is equal:

 

 

 

Shear stress at point C is equal:

 

 

 

Marking the most important values shear stress looks like below:

 

IPE shear stress

 

Of course If we want get stress at other IPE 160 points we must cut section in the mind at that points, and make calculations like above.

 

3. Shear acc. to EN 1993-1-1

 

In the standard EN 1993-1-1 in chapter 6.2.6 (5) there is condition concerning using mean shear stress. Shear stress can be calculated accordig to equationśrednie naprężenie styczneon condition that the area of one flange divided by the area of the web is bigger or equal 0.6.

 

 

Actually is availabe free program for shear stress calculation. In actual v1.0 version program calculates mean shear stress. In the future also shear stress acc. to Żurawski equation will be calculated. It is simple program for shear stress calculation only, so it is dedicated especially for designers to preliminary profile selection.