Load-Exertion Tables And Their Use For Planning – Part 1

Epley’s Equation

Prescribing strength training is indeed tricky. Things tend to vary across lifts/movements, individuals (preferences, responses, context, and so forth), gender, rep ranges, types of programs (e.g., more extensive or more intensive), splits, and so forth. But we are not completely clueless – we utilize multiple models and heuristics (rules of thumb), individual perceived feedback (e.g., rate-of-perceived-exertion (RPE), or perceived reps-in-reserve (pRIR)), as well as technology (e.g., linear-position-transducers (LPT) for the more novel velocity-based training (VBT)) to make prescription more precise.

The most common model utilized in strength training is Epley’s equation (Equation 1). Epley’s equation is a simple model that maps the basic phenomena of strength training: the higher the weight on the bar, the lower the maximum number of repetitions that can be performed.

\[\begin{equation}
1RM = (Reps \times Weight \times 0.0333) + Weight
\end{equation}\]

Equation 1

In this form (Equation 1), Epley’s equation is utilized for predicting one-repetition-maximum (1RM) (the highest weight that can be lifted for single repetition) from repetitions-to-failure. For example, if I can lift 100kg for maximum of 5 repetitions, my estimated 1RM (the weight I can lift once) is (5 * 100 * 0.0333) + 100 or 116.65 kg. This weight represents my 5RM, or five repetitions maximum.

Variation of the Equation 1 can be used for mapping n-repetition-maximums (nRMs) with %1RM (Equation 2). For example, if I have done a maximum of 5 repetitions, the estimated %1RM is 85.7%. This is of course given Epley’s model, but as always, things tend to vary. But still, Epley’s equation is a good enough (i.e., satisficing) model to be used.

\[\begin{equation}
\%1RM = \frac{1}{0.0333 \times Reps + 1}
\end{equation}\]

Equation 2

Another variation of Epley’s equation is used to estimate nRM from %1RM utilized (Equation 3). In other words, answering what the highest number of repetitions possible (nRM) is, estimated given the model when lifting a certain %1RM weight. For example, if I know my 1RM is 120kg and I am using 90kg (which is 90 / 120, or 75%), using Epley’s equation I can estimate that I can probably bang 10 reps to failure.

\[\begin{equation}
nRM = \frac{15.015}{\%1RM} – 15.015 \
\end{equation}\]

Equation 3

Using Equation 2, we can create a reps-max table (Table 1). This table is very useful as a rule of thumb for making sure you are not using unreasonable percentages (given the model, of course).

Table 1: Reps-Max Table

But How “Hard” Is It? (That’s What She Said)

Although useful for mapping max reps to %1RM, it is natural to wonder how Epley’s equation can help us in prescribing strength training, especially when we know that most of the time we do not lift to failure. In Strength Training Manual (Jovanović 2020), I have provided a thorough exploration of this conundrum and will try to keep it simple in this article.

Long story short, we are trying to develop a model that helps us plan and prescribe, and hopefully understand how hard (and thus doable and repeatable) a given set is across various rep ranges. For example, is doing 5 reps at 80% 1RM harder than doing 8 reps at 70% 1RM, and how can we progress these workouts across sessions or weeks (i.e., training phase). We are also simplifying and representing very complex phenomena of how hard a given set (or even rep) feels (i.e., emotionally, expectations, body sensations, etc) and affects performance into a simple mathematical equation or model. Put differently, we want to create a way to compare the hardness of different set schemes (e.g., 5 reps at 80% vs 8 reps at 70% 1RM) and also to create a system for making some educated guesses for progressing a given prescription scheme across time (i.e., how should we progress or make harder 5×5 at 80% versus 3×10 at 70% 1RM). There is no single best way to do this – we need to use a multiple models approach (Jovanović 2020). Each model represents a simplification with assumptions and its pros and cons.

Relative Intensity Model

One approach utilized in strength training is the Relative Intensity (RI) model. With the RI model, we are simply multiplying %1RM associated with a given nRM with a coefficient of intensity (i.e., a proxy for hardness) (Equation 4). Table 2 contains the application of the RI model using Epley’s reps-max table (Table 1).

\[\begin{equation}
\%1RM = \frac{RI}{0.0333 \times Reps + 1} \
\end{equation}\]

Equation 4

Table 2: Relative Intensity Table. Values in the columns represent RI, while the numbers in cells represent estimated %1RM for a target number of reps

Here is how Relative Intensity Table 2 works. Using the previous example of 5 reps at 80% vs 8 reps at 70% 1RM, given the RI table and Epley’s equation, we can find out that RI for the 5 reps at 80% is around 92.5-95%. More precisely, we can use Equation 5 to estimate the RI of a given rep and the %1RM scheme.