# How to calculate wheel curb climbing torque

### Definition

When designing a new vehicle or a new powertrain and driveline, the ability of the vehicle to climb a curb (kerb) must be taken into account as initial requirement. Curb climbing is the ability of the vehicle to climb a curb, from standstill, with the wheel in contact with the curb. As you’ll see in the calculation below, curb climbing is very demanding in terms of force (torque) required at the wheel.

A wheel can climb a curb in three ways, depending on the force acting on the wheel:

• with driven (pushed) force: this is the case in which there is not traction force generate at the wheel, instead the wheel is pushed through the wheel hub; this is the case in which the vehicle is rear-wheel drive and the curb is climbed by the front wheel(s).
• with driving (traction) force: this is the case in which the wheel gets torque from the powertrain and generates a traction force in the contact point with the curb; for example, the vehicle is front-wheel drive and the curb is climbed with the front wheel(s).
• with pulling force: this use case is not applicable to motor vehicles, since the point of application of the pulling force is on the top of the tire; this use case is applicable to wheelchairs but it’s interesting from the mechanics point of view.

When calculating the required force (torque) at wheel for curb climb, the following assumptions are made:

• the wheel is rigid, there is no deformation in contact with the curb;
• there is no slipping between the wheel and curb;
• the entire weight force is concentrated in the contact point between wheel and curb;
• there is no normal force acting on the wheel from the ground in front of the curb;
• the weight of the vehicle is distributed evenly between the wheels during the curb climb.

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### Driven (pushed) wheels

In the case of a driven wheel, there are two forces acting on the wheel: the weight G [N] and the push force F [N]. Both forces are acting in the centre of the wheel.

Image: Curb climb diagram – pushed wheel

The contact point P, between the wheel and curb, is the pivot point of the wheel. To find out what the force F [N] needs to be to climb the curb, we need to write the torque equilibrium equation around point P.

There are two torques acting on the wheel around point P:

• the torque due to the force F, which we are going to name TF [Nm]
• the torque due to the weight G, which we are going to name TG [Nm]

For the wheel to be in equilibrium, it means that the sum of torques around point P is zero.

TF – TG = 0
(1)

TG has negative sign because it rotates in opposite direction compared with TF.

Equation (1) can be written as:

TF = TG
(2)

If we replace the torque with the product between force and lever arm, we get:

F · (r – h) = G · s
(3)

Distance s [m] can be found using Pythagoras’ theorem:

s2 = r2 – (r – h)2
(4)

where r [m] is the wheel radius and h [m] is the curb height.

After simplifications, equation (4) gives:

$s = \sqrt{2 \cdot r \cdot h – h^{2}} \tag{5}$

Replacing (5) in (3) and solving for F, gives:

F = (G · s) / (r – h)
(6)

Equation (6) gives the curb climb force for a driven (pushed) wheel. This force will need to be generated at the motor (traction) wheels.

From the denominator we can see that if the curb height is equal with the radius, the required force for climbing will be infinite. Therefore, for the wheel to be able to climb the curb, its radius needs to be bigger than the curb height.

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### Driving (traction) wheels

When the wheel(s) receive torque from the powertrain, climbing a curb is easier, since it requires less force/torque.

Image: Curb climb diagram – traction wheel

The torque T [Nm] coming from the engine (motor) will generate a traction force R [N] and a normal force N [N] in the contact point P. To find out the value of these forces, we’ll write the equilibrium equations for the forces acting in point P.

For the wheel to be in equilibrium the sum of forces in point P, in the x and y directions, needs to be zero. This translates into the following equations:

• on the x-axis
ΣFx = 0
Rx – Nx = 0
Rx = Nx
R · cos(π/2 – θ) = N · cos(θ)
R · sin(θ) = N · cos(θ)

We can write the expression of the normal force N as:

N = R · sin(θ) / cos(θ)
(7)
• on the y-axis
ΣFy = 0
Ry + Ny – G = 0
Ry + Ny = G

Which gives the expression:

R · cos(θ) + N · sin(θ) = G
(8)

Replacing (7) in (8) and solving for R gives:

R = G · cos(θ)
(9)

Replacing (9) in (7) gives the expression of N as:

N = G · sin(θ)
(10)

Equations (9) gives the traction force required to climb the curb.

The angle θ is calculated using the formula of the sine function:

sin(θ) = (r – h) / r
(11)

which gives the expression of the angle θ [rad] as:

θ = arcsin(1 – h/r)
(12)

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### Pulled wheel

This use case does not apply to road vehicles, since the force to move the wheel is applied on the top of the wheel. This method applies to wheelchairs, where a person pushes the wheel with their hands. From the mechanics point of view it is still interesting to discuss and understand it.

Image: Curb climb diagram – pulled wheel

The same approach is applied as in the case of driven (pushed wheels). The only difference is that the torque TF [Nm] has the arm lever 2·r – h instead of r – h. This gives the expression of the pull force F [N] as:

F = (G · s) / (2 · r – h)
(13)

In this case, even if the curb height is bigger than the wheel radius, the curb can still be climbed. This is valid as long as there is no slip between the wheel and curb.

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### Motor (engine) torque

From the wheel force required to climb a curb F [N] and the wheel radius r [m], we can calculate the torque required to climb a curb T [Nm] as:

T = F · r
(14)

To calculate the torque of the internal combustion engine Teng [Nm] or electric motor Tmot [Nm] required to climb the curb, we need to know the gear ratio of the final drive i0 [-] and the gear ratio of the first gear i1 [-].

Teng/mot = T / (i0 · i1)
(15)

The assumption is that the curb is climbed in the first gear of the transmission, where the maximum available torque at the wheel is generated.

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### Example

For a better understanding let’s look at a calculation example. For a vehicle with the mass of 2000 kg and wheel radius of 0.355 m, calculate the required wheel force to climb a 0.14 m curb with one wheel. Calculate the wheel force for all three scenarios, pushed, driving and pulled wheel. What is the wheel torque in each scenario? What is the engine/motor torque, if the final drive ratio is 3.24 and the first gear ratio is 3.57?

Step 1. Calculate the weight force acting on the wheel. Since the curb is climbed only with one out of four wheels, the weight acting on one wheel is a quarter of the total mass of the vehicle.

G = (1/4) · m · g = 0.25 · 2000 · 9.81 = 4905 N

Step 2. Calculate the distance s [m] using equation (5).

$s = \sqrt{2 \cdot r \cdot h – h^{2}} = \sqrt{2 \cdot 0.355 \cdot 0.14 – 0.14^{2}} = 0.2825 \text{ m}$

Step 3. Calculate the angle θ [rad] using equation (12).

θ = arcsin(1 – h/r) = arcsin(1 – 0.14/0.355) = 0.6506 rad

We can transform from radians to degrees by multiplying the radians value with 180 and dividing to π.

θ = 0.6506 · 180 / π = 37.27 °

Step 4. Calculate the force for the driven (pushed) wheel using equations (6).

F = (G · s) / (r – h) = (4905 · 0.2825) / (0.355 – 0.14) = 6445 N

Step 5. Calculate the force for the driving (motor) wheel using equation (9).

R = G · cos(θ) = 4905 · cos(0.6506) = 3903 N

Step 6. Calculate the force for the pulled wheel using equation (13).

F = (G · s) / (2 · r – h) = (4905 · 0.2825) / (2 · 0.355 – 0.14) = 2431 N

Step 7. Calculate the wheel torque for each use case using equation (14).

Tpushed = F · r = 6445 · 0.355 = 2288 Nm
Ttraction = R · r = 3903 · 0.355 = 1386 Nm
Tpulled = F · r = 2431· 0.355 = 863 Nm

Step 8. Calculate the engine/motor torque for each use case using equation (15).

Teng/mot = Tpushed / (i0 · i1) = 2288 / (3.24 · 3.57) = 198 Nm (pushed wheel)
Teng/mot = Ttraction / (i0 · i1) = 1386 / (3.24 · 3.57) = 120 Nm (traction wheel)
Teng/mot = Tpulled / (i0 · i1) = 863 / (3.24 · 3.57) = 75 Nm (pulled wheel)

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### Calculator

In order to verify the results, you can also try the calculator below.

 r [m] = h [m] = m [kg] = i0 [-] = i1 [-] = Climb force for driven / driving / pulled wheel(s) [N] = / / Climb torque for driven / driving / pulled wheel(s) [Nm] = / / Engine/motor torque for driven / driving / pulled wheel(s) [Nm] = / / Additional parameters G [N] = s [m] = θ [°] =

Check the results for the curb height being equal with the wheel radius. Also, see what happens if the curb height is zero.