Magnel Diagram for Prestressed Concrete Design.

This article explains the importance and application of magnel diagram in prestressed concrete design.

Why Magnel Diagram?

In a previous article on verification of stress in beams published here (which will be referenced as “the previous article” henceforth), we verify whether the stress in the designed beam is within the stress limit. Again, you should note that what was done was verification of stress in the already designed beam and not the proper design itself. The design process on the other hand requires that a prestressed force (P) is prescribed for a section of a member at a particular eccentricity (e). Subsequently, this section – it could be the end, mid or whatever section in the member – can then be verified to check whether the stress in it is within the stipulated stress limit both at transfer stage and at service stage as done in the previous article.

Conversely, if we were to design (ie: prescribe P and e) for the end section of the beam in the previous article for example. We will have to go through these steps:

At transfer stage we would:

• Equate the equation for combined stress (you can find the equations in the previous article) at the top of the beam to the desired tensile stress limit at the transfer stage
• Recast the equation to be in the form y = mx + c from step 1 (where y = P and x = e.)
• Equate the equation for the combined stress at the bottom of the beam to the desired compressive stress limit at transfer stage
• Recast the equation to be in the form y = mx + c from Step 3. (Let’s assume this is equation is equ (2) and the equation in step 2 is equ (1))

If you are a math geek as most readers of this article are expected to be, you would easily say: “Solve equations (1) and (2) simultaneously to get P and e!”.

Yeah, that would have been a straightforward stuff if it were that we would not have to repeat this same process for the service stage severally. And unfortunately, we cannot prescribe differently, for the service stage, another prestressed force (P) and eccentricity (e) for the same section of a single member!

Therefore, we need a method that will allow us to prescribe P and e for a section such that it is adequate for both the transfer stage and the service stage, and that’s where the magnel diagram comes to the rescue.

The Magnel Diagram

Magnel diagram is a graphical method of optimizing the process of determining prestressing force (P) and eccentricity (e) such that the internal stresses within the particular section of the member falls within the limiting stress at both transfer and service stages. It is used to show range of possible values of prestress forces and eccentricities for a given section of a structural member. It consists of different straight lines that defines the boundary of stress condition as shown below, this region within the boundary is the feasible area for P and e.  Any combination of P and e that falls within the feasible area will satisfy the stress limit for both stages of prestress.

 

 

STEPS to Constructing a Magnel Diagram

1. Construct the stress limit equation for the section
2. Rearrange the equation to make 1/P a function of x such that the expression becomes an equation of a straight-line y = mx + c (where: 1/P = y and x = e)
3. Plot the equations of the straight lines on a graph.
4. Draw the line to represent the maximum eccentricity possible base on member section restriction
5. Choose the appropriate value for P and e within the feasible region of the graph

Recommended value for P and e?

It should be noted that it is highly recommended that the value of 1/P is large so that a lower prestress force is applied on the member, and this is compensated for with high value of eccentricity (e).

The advocacy for lower prestressing force is due to the following factors

• Economy: Lower prestressing for necessitates lesser ring reinforcement behind the bearing plate during prestressing operation as the member is less prone to bursting. The tendons that would be required for the prestressing of operation will be lesser compared to when the prestressing force to be applied is large.

 

• Ease of prestressing Operation: When the prestressing force is small this translates to the need for smaller jack and fewer tendons for the prestressing operation which eventually makes the operation less hassle free.

 

Main factor affecting the eccentricity of tendons

 

The eccentricity of tendons is mainly affected by the sectional limitation of the member such as the need to provide adequate cover to the tendon and space for the provision of shear links.

 

When Higher Prestressing force than sufficient is desired?

High prestressing force than sufficient might be considered sometimes with the aim of increasing the shear capacity of the member so that lesser shear links than would have been otherwise required can be used. It can also be considered to make the member much more compact and resistant to defection.

Equations for Magnel diagram:

The below four equations can be used to construct a magnel diagram at any section of a beam.

At Transfer

$
\frac{1}{P}\,\,\geqslant \,\,\frac{\left( -\frac{1}{A}\,\,+\,\,\frac{e}{Z_t} \right)}{\frac{\eta \,\,M_{\min}}{\gamma _{sup\,\,}Z_t}+\frac{\eta \,\,f_{tt}}{\gamma _{sup}}}
$

$
\frac{1}{P}\,\,\geqslant \,\,\frac{\left( -\frac{1}{A}\,\,-\,\,\frac{e}{Z_b} \right)}{\frac{\eta \,\,M_{\min}}{\gamma _{sup\,\,}Z_b}+\frac{\eta \,\,f_{tc}}{\gamma _{sup}}}
$

At Service

$
\frac{1}{P}\,\,\geqslant \,\,\frac{\left( -\frac{1}{A}\,\,+\,\,\frac{e}{Z_t} \right)}{\frac{\,\,M_{\max}}{\gamma _{inf\,\,}Z_t}+\frac{\eta \,\,f_{sc}}{\gamma _{inf}}}
$

$
\frac{1}{P}\,\,\leqslant \,\,\frac{\left( -\frac{1}{A}\,\,-\,\,\frac{e}{Z_b} \right)}{\frac{\,-M_{\max}}{\gamma _{inf\,\,}Z_b}+\frac{\eta \,\,f_{st}}{\gamma _{inf}}}
$

Click here to study a worked example on how these equations are used in constructing a magnel diagram.

 

Here are the definitions of the parameters in the equations:

P = prestressing force at service (after long-term loss)

M_min = Moment due to the self-weight of the member

M_max = Moment due to load at service on the member

η= (1- % loss at service)/ (1- % loss at transfer)

γ_min= Inferior load factor

γ_sup= Superior load factor

e = eccentricity of the tendons

Z_t= Section moduli to the top fiber

Z_b= Section moduli to the bottom fiber

A = Cross-sectional area of the member

f_tt = Limiting tensile stress at transfer

f_tc = Limiting compressive stress at transfer

f_sc = Limiting compressive stress at service

f_st = Limiting tensile stress at service.

 

Methodology of member design?

It should be noted that the magnel diagram is often used to design a single section in the entire span of a member. The mid-span is typically singled out for design in a simply supported beam as it is subjected to the highest external bending stress thereby requiring the highest number of prestressed tendons. Subsequently, debonding technique can be employed in a pre-tensioned member to render some of the tendons in-active in other sections which are subjected to lesser external stress so that the prestress is varied along the member.

As for a post-tensioned member, the method of varying the eccentricity across the length of the member so that the eccentric stress due to prestress is applied proportionately to counteract the external bending stress acting on the member comes in handy. The eccentricity is high at section subjected to high external stress, and low at sections with low external stress. This is done by estimating a suitable cable profile throughout the member to be designed.