Calculating Thrust: How Many Pushing Bags Do You Need?

A quarry foreman calculating block dimensions and marking pushing bag insertion points on a massive cut granite block
Engineering precision: Toppling a massive block is not about brute force; it is about exploiting physics, leverage, and the block’s center of gravity.

In the quarrying industry, many operators purchase pushing bags based solely on a “gut feeling” or a fundamental misunderstanding of physics, leading to either failed extraction attempts or vastly overspending on unnecessarily massive equipment. According to “Rock Mass Overturning Stability” principles published by the International Society for Rock Mechanics (ISRM), toppling a monolithic stone block is an exercise in Statics and Torque (Moment), not raw lateral displacement. You do not need 300 tons of direct linear force to tip a 300-ton block. You simply need to apply enough leverage to push its Center of Gravity past its tipping point.

A large-scale granite quarry in South Africa recently encountered this exact engineering dilemma. After successfully utilizing a wire saw to separate a colossal 400-ton base block, the management team assumed they needed to purchase two of MosCut’s largest MCBL3030 bags (450 tons each) just to move it. This assumption far exceeded their monthly tool budget. MosCut field engineers intervened and introduced them to the “Tipping Moment Formula.” By mathematically analyzing the block’s geometry, the engineers instructed the crew to position two significantly smaller MCBL1020 bags (100 tons each) much higher up in the wire saw cut, drastically increasing their leverage. The 400-ton block was safely and effortlessly toppled using only an actual applied thrust of 180 tons, saving the quarry 40% on their equipment procurement costs.

The Physics of Toppling

You do not need 300 tons of thrust to push a 300-ton block. You only need to manipulate leverage.

Imagine trying to tip over a tall, heavy refrigerator. If you place your hands at the very bottom near the floor and push, it takes an immense amount of strength. However, if you place your hands at the very top of the refrigerator and push, it tips over with minimal effort. This is the law of the lever.

The exact same physics apply to a 10-meter-tall marble block. The stone pivots on its bottom front edge (the fulcrum). The higher you place your MosCut pneumatic pushing bag within the wire saw cut, the longer your “lever arm” becomes, and the less actual tonnage you need to overcome the block’s weight. Therefore, the required capacity of your pushing bags is entirely dependent on where you place them.

Diagram illustrating the physics of leverage: pushing higher on a block requires significantly less force to tip it over the fulcrum
Leverage dictates capacity: Placing a pushing cushion higher up the cut drastically reduces the required thrust capacity.

Step 1: Calculate the Block Weight

Before you can move the mountain, you must know exactly what it weighs.

To determine the weight of the separated block, you must multiply its volume by the specific density of the geological material. Use the following formula:

$$W = L times B times H times rho$$

Where:

  • $W$ = Total Weight of the block (in Tons)
  • $L$ = Length of the block (in Meters)
  • $B$ = Breadth/Thickness of the block (in Meters – depth from front face to the back cut)
  • $H$ = Height of the block (in Meters)
  • $rho$ = Density of the rock (in Tons per Cubic Meter)
Geological MaterialAverage Density ($rho$)
Sandstone / Travertine2.3 – 2.4 t/m³
Limestone2.5 – 2.6 t/m³
Marble2.6 – 2.7 t/m³
Granite / Quartzite2.8 – 3.0 t/m³
Basalt / Gabbro2.9 – 3.1 t/m³

Step 2: The Tipping Moment Formula

Leverage is your ultimate weapon. Where you place the bag dictates how much power you need.

Once you have calculated the block’s total weight ($W$), you can determine the exact amount of pushing force required to tip it over based on where you place your pushing bag. We use the static equilibrium formula:

$$F_{required} = frac{W times (B / 2)}{H_{bag}}$$

Where:

  • $F_{required}$ = The absolute minimum thrust required to tip the block (in Tons).
  • $B / 2$ = Half the breadth/thickness of the block (this is the distance from the pivot edge to the block’s center of gravity).
  • $H_{bag}$ = The height from the bottom of the block to the center of the pushing bag.
📝 Real-World Example:
You have a Marble block weighing 200 Tons ($W = 200$). The block is 2 meters thick ($B = 2$). You insert the pushing bag 1 meter off the quarry floor ($H_{bag} = 1$).
$$F_{required} = frac{200 times (2 / 2)}{1} = 200 text{ Tons of thrust required.}$$ However, if you slide the exact same bag higher up the cut, to 2 meters off the floor ($H_{bag} = 2$):
$$F_{required} = frac{200 times (2 / 2)}{2} = 100 text{ Tons of thrust required.}$$ Conclusion: By placing the bag higher, you cut the required pushing force in half!

Step 3: Sizing the Bags & The Expansion Drop-off

Bag capacity is not static. It is a calculation of surface area multiplied by internal air pressure.

When you look at a MosCut MCBL1020 bag rated for 100 Tons, that rating is based on the bag being inflated at an optimal 7-8 Bar of air pressure while completely flat. However, as the bag inflates to push the stone, it begins to take on a rounded, “balloon-like” shape.

The Expansion Drop-off: As the bag becomes rounder, the actual surface area making physical contact with the flat rock face decreases. Because $Thrust = Area times Pressure$, less contact area means less pushing force. Therefore, you must never buy a bag that exactly matches your $F_{required}$. You must always add a 20% to 30% Safety Margin to account for thrust decay during the expansion stroke. If your formula dictates you need 150 tons of force, you should equip your operation with at least 200 tons of rated bag capacity.

Cross-section diagram showing how a pushing bag's flat contact area decreases as it inflates, reducing overall pushing force
Thrust decay: As the cushion expands and rounds out, the effective contact area against the stone decreases. Always calculate a 30% safety margin.

⚠️ The “Rule of Two”: Why Multiple Bags are Mandatory

Even if you calculate that a single MosCut MCBL2020 (200-ton capacity) bag provides more than enough force to tip your block, you must never use a single bag placed in the center of the block. A single point of pressure creates a central pivot point. If the block has hidden fractures, uneven density, or if the floor is slightly sloped, the block will not fall straight forward. Instead, it will undergo a catastrophic “Twisting” or “Spinning” motion as it falls sideways off the pivot point. This unpredictable spin can destroy the block, damage neighboring stone, and crush personnel. You must always adhere to the Rule of Two: Utilize a minimum of two identical pushing bags placed symmetrically (roughly 1/4 of the total length inward from each edge). This provides two synchronized points of leverage, forcing the block to tip forward in a perfectly straight, controlled line.

Equip Your Quarry with Engineered Precision

Now that you understand the mechanics of safe block toppling, equip your crew with the industry’s most reliable tools. View MosCut’s full range of Kevlar-reinforced pushing bags to match your exact extraction parameters.

View MosCut Pushing Bags

Advanced Engineering FAQ

1. What happens if my air compressor cannot reach 8 Bar?
The rated tonnage of a pushing bag is directly proportional to the supplied pressure. If your compressor only reaches 4 Bar (roughly half the optimal pressure), the pushing bag will only output approximately 50% of its rated pushing capacity. You must recalculate your required bag size accordingly.
2. The math says I can place the bag at the very top, but will it push far enough?
This is the “Leverage Trade-off.” Placing the bag higher up requires less force, but it requires more expansion distance (stroke length) to push the center of gravity past the tipping point. If placed too high, the bag may fully inflate to 300mm wide without the block tipping. Often, the middle of the cut is the optimal balance between force and expansion distance.
3. How do I calculate the weight of a block on a sloped quarry bench?
If the block is sitting on a downward-sloping bench, gravity is already assisting you. The required tipping force ($F_{required}$) is reduced significantly. Conversely, if pushing uphill, you must drastically increase your pushing capacity. These complex scenarios often require specialized trigonometric calculations by the quarry engineer.
4. Why did the block slide backward instead of tipping forward?
If a block slides rather than tipping, it means the pushing force exceeded the friction coefficient between the bottom of the block and the bedrock floor before the tipping moment was reached. This usually happens if the block is very wide (large $B$) and very short (small $H$). Pushing higher up ($H_{bag}$) helps prevent this.
5. Should I use different sized bags on the same block?
No. When using the “Rule of Two,” you should always use identical bags (e.g., two MCBL1015s) connected to the same control manifold. Mismatched bags will expand at different rates and apply uneven pressure, leading to the dangerous twisting motion we want to avoid.
6. Does the width of the wire saw cut affect the pushing force?
Yes, due to the Expansion Drop-off. If the initial cut is excessively wide (e.g., 50mm due to rock shifting), the bag is already partially expanded and rounded before it even touches both sides. It will deliver significantly less thrust than if inserted into a tight 10mm gap. Use flat steel spacer plates to fill wide gaps.
7. Can I stack pushing bags to double the pushing force?
No. Stacking bags vertically (one in front of the other within the same gap) does not double the pushing force. It only doubles the expansion distance. To double the pushing force, you must place the bags side-by-side along the length of the cut.
8. How do I account for uncut rock at the bottom of the block?
You cannot topple a block that is not fully separated. If even a small bridge of solid rock remains uncut at the bottom corners, its tensile strength will easily resist hundreds of tons of pushing force. The block must be 100% cleanly cut by the wire saw or splitters before inflating the bags.
9. What is the maximum safe expansion width of a pushing bag?
Most standard MosCut bags safely expand to a maximum thickness of 250mm to 350mm depending on the model. Never attempt to over-inflate a bag beyond its operational limit in an attempt to push a block further; this risks a seam blowout.
10. Is water pressure (Hydro) calculated differently than air pressure?
The mathematical formula for the required thrust ($F_{required}$) remains exactly the same regardless of the medium. However, because water is incompressible, hydrostatic bags can be safely subjected to much higher internal pressures (e.g., up to 20 Bar with specialized pumps), yielding massively higher final thrust capacities from the same surface area.