STRENGTH CALCULATION OF THE CRANK-CONNECTING ROD MECHANISM PARTS

Cylinder Liner Calculation

Cylinder liners are among the most heavily loaded engine parts. They experience stresses from combustion gas forces, piston side thrust, and thermal loads. The harsh operating conditions require manufacturing liners from high-quality alloyed cast irons (SCh28-48, SCh35-56) or steel. The strength calculation only accounts for the primary loads: maximum gas pressure, piston side pressure, and the temperature gradient across the liner wall. The most critical stress is caused by the maximum combustion pressure Pz, which stretches the cylinder along its generatrix and around its ring cross-section (fig. 5.1)

The main structural dimensions of the liner are chosen to provide the strength and stiffness needed to prevent ovalization during engine assembly and operation. The wall thickness, initially chosen by design convention, is then verified using the formula for cylindrical sleeves.

Calculated liner wall thickness, m

σ_z:

allowable tensile stress for cast iron, MPa

p_z:

maximum combustion pressure, MPa;

δ_r:

cylinder liner wall thickness of the prototype engine, m

cylinder diameter, m

Tensile stress in the liner from maximum gas pressure, MPa

σ_p:

Thermal stress in the liner, MPa

ΔT

temperature difference between the inner and outer liner surface, K.

modulus of elasticity of cast iron, MPa;

α_c

coefficient of linear expansion;

Poisson's ratio.

σ_t:

Combined stress in the liner from gas pressure and temperature difference, MPa.

σ_max

allowable combined stress, MPa (cast iron: 100–130 MPa; steel: 180–200 MPa)

Outer surface

σ_∑`:

combined stress (tension + thermal), MPa

Inner surface

σ_∑``:

combined stress (tension − thermal), MPa

Cylinder Head Stud Calculation

Load-bearing studs are used to fasten the cylinder head to the block/crankcase. Their condition is affected by the pre-tightening force, gas pressure, and the load arising from temperature differences and the coefficient of linear expansion of the head, block/crankcase, and cylinder materials. The number of studs, their pre-tightening, and their structural dimensions must ensure the head seats securely against the block. The stud calculation scheme is shown in fig. 5.2.

Studs are manufactured from high-elasticity carbon steels or high-alloy steels (18KhNMA, 18KhNVA, 40KhNMA).

When the engine is cold and not running, the studs are loaded by the pre-tightening force, MN, determined by the formula

stud tightening coefficient;

basic load coefficient of the threaded joint; empirically χ = 0.15–0.25, generally decreasing as the bolt diameter decreases.

i_shp

number of studs per cylinder;

d_shp

stud diameter, m

t_shp

stud thread pitch, m

d_v

inner diameter of the stud, m

Fk/Fp

projected combustion chamber surface area onto the plane perpendicular to the cylinder axis, m²; Fk/Fp ≈ 1.1

F_p

F_k

P_zmax`

the combustion gas pressure force acting on a single stud, MN. Determined by the formula

P_p

the pre-tightening force, MN,

P_pmax

Total force stretching the stud, MN

The maximum and minimum stresses, MPa, arising in the stud are determined at the stud section, accounting for the internal thread diameter

σ_max

σ_min

Mean stress and cycle amplitude, MPa

σ_m

σ_a

Stud safety factor must be greater than 2

σ_-1p

endurance limit in tension-compression, MPa;

K_σ

stress concentration coefficient in the thread;

coefficient accounting for surface finish quality.

n_shp

Piston Group Calculation

Piston Calculation

The piston, shown schematically in fig. 5.3, is the most heavily loaded element of the piston group: it withstands high gas, inertial, and thermal loads, so its material is held to elevated requirements. It is mostly made from aluminum alloys, rarely from cast iron.

The verification calculation of the piston elements is performed without accounting for variable loads, whose magnitude is instead reflected in the corresponding allowable stresses. The crown, head wall, top ring land, bearing surface, and skirt of the piston are calculated.

d_m

oil drain hole diameter, m;

δ_p

piston crown wall thickness, m;

ring radial thickness, m;

Δ_t

ring radial clearance in the piston groove, m;

piston head wall thickness, m.

r_i

inner radius of the piston crown, m

Bending stress in the piston crown, MPa

σ_{iz_max}

maximum allowable bending stress, MPa

σ_iz

Compressive stress at section x-x, MPa

P_zd

maximum gas pressure force on the piston crown, MN

n_m`

number of oil channels in the piston;

d_k

piston diameter at the groove bottoms, m

d_i

inner diameter of the piston, m

F_m`

longitudinal cross-sectional area of the oil channel, m²

F_x-x

cross-sectional area at x-x, m²

σ_{szh_max}

maximum allowable compressive stress, MPa

σ_szh

Tensile (rupture) stress at section x-x, MPa

piston stroke, m

connecting rod length, m

m_p

mass of the piston group, kg;

n_max

crankshaft rotation speed at idle, min⁻¹;

m_x-x

mass of the piston head above section x-x, kg

ω_x-x

maximum angular velocity, s⁻¹

crank radius, m

kinematic parameter of the crank-connecting rod mechanism;

P_j

inertia force of the reciprocating masses, N

σ_{r_max}

maximum allowable tensile stress, MPa

σ_r

Stresses in the top ring land:

h_p

Top ring land thickness, m

shear stress, MPa

σ_iz

bending stress, MPa

σ_{∑ max}

maximum allowable combined stress, MPa

σ_∑

combined bending/shear (equivalent) stress, MPa

Maximum specific pressure, MPa, of the piston skirt (h_ubk = 0.053 m) and side surface (H_ubk = 0.076 m) against the cylinder wall

h_ubk

piston skirt

side surface

N_max

maximum normal force at a crank angle of 380°

q_{1 max}

for modern automotive and tractor engines, q1 = 0.3–1.0 MPa

q₁

q_{2 max}

for modern automotive and tractor engines, q2 = 0.2–0.7 MPa

q₂

Piston head and skirt diameters, m

To prevent piston seizure during engine operation, the head diameter Dg and skirt diameter Du are determined based on the necessary clearances Δg and Δu between the cylinder and piston walls in the cold state. Based on statistical data, for aluminum pistons with solid (non-slotted) skirts: Δg = (0.006–0.008)·D and Δu = (0.001–0.002)·D; for cast iron pistons: Δg = (0.004–0.006)·D and Δu = (0.001–0.002)·D. Once Δg and Δu are determined, the following dimensions are calculated

α_c

coefficient of linear expansion of the piston material

T_c

cylinder wall temperature;

T_iu

piston skirt temperature;

T_g

piston head temperature;

T₀

initial temperature of the cylinder and piston, K

Δ_g

initial clearance at the piston head

Δ_iu

initial clearance at the piston skirt

D_g

Piston head diameter, m

D_iu

Piston skirt diameter, m

Δ_g`

calculated clearance at the piston head

Δ_iu`

calculated clearance at the piston skirt

Piston Ring Calculation

A_{0 coef}

A0 coefficient

ring radial thickness, m;

modulus of elasticity of the ring material (gray cast iron: E = 1.0×10&sup5; MPa; alloyed cast iron: E = 1.2×10&sup5; MPa; steel: E = (2.0–2.3)×10&sup5; MPa) [3].

A₀

difference between the ring gap clearance in the free and installed (working) states;

p_cp

Average ring pressure on the cylinder wall, MPa

coefficient depending on the ring installation method [3].

σ_{iz_max}

maximum allowable bending stress

σ_iz1

Ring bending stress in the working state, MPa

σ_iz2

Ring bending stress when installing on the piston, MPa

Ring Pressure on the Cylinder Wall

ψ, °	0	30	60	90	120	150	180
μ_k	1.05	1.05	1.14	0.90	0.45	0.67	2.85
p, MPa

Piston Pin Calculation

coefficient accounting for the piston pin mass. For carburetor engines k = 0.76–0.86, for diesel engines k = 0.68–0.81 [3].

P_j

inertia force of the piston group, MN

Calculated force acting on the piston pin, MN

d_p

outer diameter of the pin, m;

l_sh

length of the connecting rod bushing, m.

q_{sh_max}

for modern automotive and tractor engines, q_sh = 20–60 MPa

q_sh

Specific pressure of the pin on the connecting rod's small-end bushing, MPa

l_p

pin length, m;

b_p

distance between the boss end faces, m.

q_{b_max}

for modern automotive and tractor engines, q_b = 15–50 MPa

q_b

Specific pressure of the pin on the piston bosses, MPa

d_pv

inner diameter of the piston pin, m

ratio of the pin's inner to outer diameter

σ_{iz_max}

For automotive and tractor engines [σ_bend] = 100–250 MPa [3].

σ_iz

Bending stress at the mid-section of the pin, MPa

θ_max

For automotive and tractor engines [τ] = 60–250 MPa [3]. The lower limits apply to tractor engines, and the upper limits to pins made from alloyed steel [3].

Shear stress in the sections between the bosses and the connecting rod head, MPa

modulus of elasticity for steel, MPa;

Δd_{p_max}

The maximum pin ovalization must not exceed 0.02–0.05 mm [3].

Δd_p

Largest increase of the pin's horizontal diameter due to ovalization, m

σ_max

maximum allowable ovalization stress

Ovalization stress on the outer surface of the pin, MPa:

σ_α0

in the horizontal plane (point 1, at α = 0°)

σ_α90

in the vertical plane (point 3, at α = 90°):

Ovalization stress on the inner surface of the pin, MPa:

σ_αi0

in the horizontal plane (point 2, at α = 0°)

σ_αi90

in the vertical plane (point 4, at α = 90°):

Connecting Rod Group Calculation

Small (Piston Pin) End Calculation

The calculated elements of the connecting rod group are: the small end, the big end, the rod shaft, and the connecting rod bolts. Connecting rods are manufactured from steel with high strength and yield limits (40Kh, 18KhNVA, 49KhNMA).

The small end of the connecting rod is checked for tensile failure at section I-I (fig. 5.6) under the inertia forces of the reciprocating masses of the piston assembly and the small end itself.

m_sh

mass of the connecting rod

k_msh

(0.06–0.09);

d_gn

outer diameter of the small end, m;

m_vg

(0.06–0.09)·m_sh — mass of the small-end portion above section I-I, kg;

h_g

(d_gn − d_p)/2 — radial wall thickness of the small end, m;

σ_max

Tensile stress must not exceed 35 MPa [3].

Tensile (rupture) stress

Big (Crankpin) End Calculation

m_shp

reciprocating (translating) mass portion of the connecting rod group, kg.

m_shk

rotating mass portion of the connecting rod group, kg.

m_kr

mass of the big-end cap, kg (≈0.25 of the connecting rod mass)

d_shsh

crank pin journal diameter, m;

t_v

bearing shell wall thickness, m;

l_k

big-end length, m.

c_b

distance between the connecting rod bolts, m;

F_g

combined cross-sectional area of the cap and bearing shell at the calculated section, m²

I_v

moment of inertia of the bearing shell, m⁴

r_i

inner radius of the connecting rod big end, m

I_k

moment of inertia of the cap, m⁴

W_iz

section modulus of the calculated cap cross-section, excluding stiffening ribs, m³

P_p

Maximum rupture force, MN

σ_{iz_max}

maximum allowable stress, MPa

σ_iz

Bending stress of the cap and bearing shell, MPa

Connecting Rod Shaft Calculation

The connecting rod shaft is checked for fatigue strength under the alternating combined forces that arise during engine operation.

h_sh

b_sh

a_sh

t_sh

F_cp

cross-sectional area at section B-B, m²

P_szh

compression from the force P_comp = P_max, MN, taken from the dynamic calculation.

P_p

tension from the force P_tens = -P_jmax, MN, taken from the dynamic calculation.

σ_czh

Stress from the compressive force, MPa

σ_r

Stress from the tensile force, MPa

σ_cp

Mean stress over the cycle, MPa:

σ_a

Stress amplitude over the cycle, MPa:

σ_-1p

endurance limit in tension-compression, MPa.

ε_σ

coefficient accounting for surface finish quality;

α₁

coefficient depending on the material characteristics;

Safety factor at section B-B, determined from the fatigue limit (must be greater than 2.5)

Connecting Rod Bolts Calculation

Connecting rod bolts must have high strength and reliability. They are manufactured from steel grades 35Kh, 40Kh, 45KhMA.

i_b

number of connecting rod bolts.

basic load coefficient of the threaded joint.

d_shb

nominal bolt diameter, m;

t_shb

thread pitch, m;

d_v

inner (root) diameter of the bolt thread, m

P_pr

Pre-tightening force, MN

P_b

Total force stretching the bolts, MN

σ_max

Maximum stress in the bolt, MPa

σ_min

Minimum stress in the bolt, MPa

σ_m

Mean cycle stress, MPa

σ_a

Cycle stress amplitude, MPa

σ_-1p

endurance limit in tension-compression, MPa.

K_σ

stress concentration coefficient in the thread;

coefficient accounting for surface finish quality;

n_b

Bolt safety factor must be greater than 2

Crankshaft Group Calculation

The crankshaft is one of the most complex parts, operating under variable forces and moments.

A simplified crankshaft calculation is performed to determine the specific pressure on the crank pin journal, as well as the torsional safety factor of the main and crank pin journals, and the stress in the crank pin journal and cheek from the maximum forces K and T.

Example input data: forces K = 38.52 kN, T = 8.94 kN; crank pin journal diameters d_shsh = 0.053 m (outer), d_shsh_vn = 0.027 m (inner); arm lengths L_kv = 0.091 m, l_kv_1 = 0.046 m, l_kv_2 = 0.046 m; a_kv_1 = 0.017 m; cheek width and height b_kv_sch = 0.023 m, h_kv_sch = 0.046 m; journal overlap X_kv = 0.024 m; crank radius R = 0.04 m; shaft material — steel 45Kh.

l₁

l₂

d_shshvn

inner diameter of the crank pin journal, m.

σ_max

maximum allowable combined stress, MPa

Determine the support reactions,

R_K₁

R_K₂

R_T₁

R_T₂

Crank Pin Journal Calculation

M_ik

bending moment, MN·m

W_u

section modulus in bending, m³

σ_ik

Bending stress in the crank plane, MPa

M_it

bending moment, MN·m

σ_it

Bending stress in the plane perpendicular to the crank, MPa

σ_iz

Combined bending stress, MPa

M_kr

torque, MN·m

W_u

section modulus in torsion, m³

Torsional stress, MPa

Equivalent (reduced) stress, MPa

Crankshaft Cheek Calculation

The cheek experiences compressive or tensile stress from the force K, bending stress in the crank plane from the force K, bending stress in the plane perpendicular to the crank from the force T, and torsional stress from the force T.

a₁

a₂

journal overlap, m

cheek width, m

cheek height, m

Compressive (tensile) stress,

σ_szh

MPa

Bending stress in the crank plane from the force K,

M_K1

MN·m

W_ik

section modulus in bending, m³

σ_{izg max}

Steel: 120–180 MPa, Cast iron: 80–100 MPa [6]

σ_K1

MPa

Bending stress in the plane perpendicular to the crank from the force T,

M_T1

MN·m

W_it

section modulus in bending, m³

σ_it

MPa

Torsional stress from the force T,

M_kr

MN·m

coefficient (Table 4.1, depends on h/b)

W_kr

section modulus in torsion, m³

τ_max

Steel: 120–180 MPa, Cast iron: 80–100 MPa [6]

τ_max

on the wide side of the cheek, MPa

τ_min

on the narrow side of the cheek, MPa