حاسبة انحراف رافعة الجيب

This free online رافعة ذراع deflection calculator estimates the total downward tip deflection of a cantilever jib beam under a lifted load, combining the bending of the horizontal cantilever arm and the lateral sway induced by column flexure. Enter the rated capacity, hoist weight, jib radius, column height, section properties, and safety factor — and get the calculated total deflection δ in millimetres instantly.

The result is compared against the allowable deflection limits specified in ISO, FEM, and GB/T standards to confirm compliance. Excessive deflection causes load swinging, positioning inaccuracy, and permanent structural deformation. A significant gap between the calculated and observed deflection during inspection indicates overload, loose connection bolts, foundation movement, or structural damage — making this calculator a useful diagnostic tool as well as a design aid.

Applicable crane types: free-standing pillar jib cranes, FEM standard jib cranes, and workstation jib cranes.

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Deflection Formula

Total deflection δ combines two components: deflection at the tip of the cantilever beam due to beam bending, and additional tip displacement caused by column lateral flexure under the same load.

\[\delta = \frac{p \cdot a^3}{3EJ_1} + \frac{p \cdot a^2 \cdot L}{EJ_2}\]

Column Section Inertia (J₂)

For a hollow circular steel pipe column:

\[J_2 = \frac{\pi}{64}(D^4 - d^4)\]

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Parameter Definitions

رمز	المعلمة	وحدة	الوصف
δ	Total deflection	مم	Downward tip displacement of the jib under load
p	Design load	N	p = k × Q + Gh (safety factor × rated capacity + hoist weight)
k	Safety factor	—	Dynamic amplification factor based on operating condition (see table below)
Q	السعة المقدرة	N	Maximum safe working load on the hook
Gh	Electric hoist weight	N	Self-weight of the hoist unit travelling on the jib beam
a	Cantilever beam effective slewing radius	مم	Horizontal distance from column centreline to hook centreline at maximum reach
L	Column effective height	مم	Effective free height of the column from foundation top to jib connection point
E	Steel elastic modulus	N/mm²	Typically 2.06 × 10⁵ N/mm² (2.06 × 10¹¹ Pa) for structural steel
J₁	Cantilever beam section inertia	سم⁴	Second moment of area of the I-beam jib section about the bending axis
J₂	Column section inertia	سم⁴	Second moment of area of the hollow pipe column about the bending axis
D	Column outer diameter	مم	Outside diameter of the steel pipe column
d	Column inner diameter	مم	Inside diameter of the steel pipe column (wall thickness = (D−d)/2)

Unit note: Q and Gh are entered in Newtons (N). If the capacity is given in tonnes (t), multiply by 9,810 to convert to N (e.g. 1 t = 9,810 N). J₁ is entered in cm⁴ as tabulated; the formula internally converts to mm⁴ (1 cm⁴ = 10,000 mm⁴).

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Safety Factor (k) Selection

The safety factor k amplifies the static load to account for dynamic effects during crane operation. Select k based on the actual operating condition:

k Value	Operating Condition	التطبيق النموذجي
1.00	Normal static operation	Slow, smooth lifts with no acceleration or braking impact
1.10	Start/stop or VFD operation (lighter)	Standard electric hoist with contactor control, moderate start/stop frequency
1.15	Start/stop or VFD operation (heavier)	Higher cycle frequency, more frequent starts under load
1.20	Faster motion or noticeable impact (lighter)	Faster hoist speeds, occasional load swing or impact at pick-up
1.30	Faster motion or noticeable impact (heavier)	High hoist speeds, regular shock loading at hook engagement

ملاحظة: k = 1.00 is suitable only for preliminary estimation under purely static conditions. For any operational jib crane, k ≥ 1.10 should be used. When in doubt, select the next higher value — underestimating the dynamic factor leads to jib designs that deflect beyond limits in service.

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How to Use the Calculator

Step 1 — Select Safety Factor (k) Choose k from the dropdown based on the crane's operating speed and control system.

Step 2 — Enter Rated Capacity Q (N) Enter the maximum safe working load in Newtons. Convert from tonnes if needed: Q(N) = capacity(t) × 9,810.

Step 3 — Enter Electric Hoist Weight Gh (N) Enter the self-weight of the hoist unit in Newtons. This is the trolley/hoist assembly weight that rides along the jib beam at all times, regardless of whether a load is on the hook. The value is available from the hoist manufacturer's data sheet.

Step 4 — Enter Cantilever Beam Effective Slewing Radius a (mm) Enter the horizontal distance in millimetres from the column centreline to the hook centreline when the trolley is at maximum outreach (tip of jib). This is the worst-case bending arm for deflection.

Step 5 — Enter Column Effective Height L (mm) Enter the effective height of the column in millimetres. For a free-standing pillar jib crane, this is the distance from the top of the foundation (or floor anchor) to the point where the jib arm connects to the column. For a wall-mounted jib, this would be the distance between the two bracket attachment points.

Step 6 — Enter Steel Elastic Modulus E (N/mm²) For standard structural steel (Q235, Q345, S235, S355, A36), E = 206,000 N/mm² (2.06 × 10⁵ N/mm²). This value is pre-filled as the default and rarely needs to change unless a non-standard steel alloy is specified.

Step 7 — Select I-Beam Specification and J₁ Select the I-beam section used for the jib cantilever arm from the dropdown. The corresponding section inertia J₁ (Ix, cm⁴) is auto-filled from the GB/T 706-2016 / ISO 657-1 hot-rolled I-beam table. For beam sizes not in the dropdown, refer to the full reference table below and enter J₁ manually.

Step 8 — Enter Column Pipe Dimensions D and d (mm) Enter the outer diameter D and inner diameter d of the hollow steel pipe column in millimetres. The calculator derives J₂ automatically from these two dimensions. Ensure D > d; the wall thickness is (D − d) / 2.

Step 9 — Calculate Click Calculate Deflection. The calculator returns the total tip deflection δ in millimetres.

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Deflection Limit References

The calculated δ must be compared against the allowable deflection for the crane's duty class and span. Common limits from international standards:

قياسي	Allowable Tip Deflection
GB/T 14405 (China)	δ ≤ a / 350
FEM 1.001 (Europe)	δ ≤ a / 350
معايير ISO (general)	δ ≤ a / 350 (typical reference)

Where a is the cantilever beam effective slewing radius (jib length). For example, a jib with a = 3,000 mm must have a calculated deflection δ ≤ 3,000 / 350 ≈ 8.6 mm to comply.

Practical note: Some procurement specifications or facility engineers apply tighter limits (a/400 or a/500) for precision assembly applications or where load positioning accuracy is critical. Always check the project specification against the calculated value, not just the standard default.

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I-Beam Section Inertia Reference Table (GB/T 706-2016 / ISO 657-1)

The following table lists the second moment of area Ix (cm⁴) about the strong axis for hot-rolled I-beams. Values are taken from Annex A, Table A.1 of GB/T 706-2016, which is essentially consistent with ISO 657-1:2026.

Built-in Dropdown Selections

I-Beam	J₁ = Ix (cm⁴)
16	1,130
18	1,660
20a	2,370
22a	3,400
25b	5,280
28a	7,110
32a	11,100

Full Reference Table (Manual Entry)

I-Beam	Ix (cm⁴)	I-Beam	Ix (cm⁴)
10	245	12	436
12.6	488	14	712
20b	2,500	22b	3,570
24a	4,570	24b	4,800
25a	5,020	27a	6,550
27b	6,870	28b	7,480
30a	8,950	30b	9,400
30c	9,850	32b	11,600
32c	12,200	36a	15,800
36b	16,500	36c	17,300
40a	21,700	40b	22,800
40c	23,900	45a	32,200
45b	33,800	45c	35,300
50a	46,500	50b	48,600
50c	50,600	55a	62,900
55b	65,600	55c	68,400
56a	65,600	56b	68,500
56c	71,400	63a	93,900
63b	98,100	63c	102,000

Source: GB/T 706-2016, Annex A, Table A.1. Ix is the moment of inertia about the x-axis (strong axis bending).

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Understanding the Two Deflection Components

The total tip deflection of a pillar jib crane is the sum of two physically distinct deformations. Understanding both helps with troubleshooting and with making the right design decisions when deflection exceeds the limit.

Component 1 — Cantilever Beam Bending: pa³ / (3EJ₁)

This is the classical cantilever beam deflection formula. The jib arm is modelled as a horizontal cantilever beam, fixed at the column connection and free at the tip. The load p (rated capacity plus hoist weight, amplified by k) acts downward at the tip. Deflection is proportional to the cube of the jib length (a³), making jib reach the most sensitive parameter — doubling the jib reach increases beam bending deflection by a factor of eight.

To reduce this component:

• Increase J₁ by selecting a larger I-beam section
• Reduce jib reach a (if operationally acceptable)
• Use a higher-grade steel with the same section (no benefit — E is virtually identical for all structural steels)

Component 2 — Column Flexure: pa²L / (EJ₂)

The point load p acting at the tip of the jib also applies a bending moment to the column equal to p × a. This moment causes the column to lean slightly forward, which displaces the jib tip by an additional amount. This component is proportional to a² and L (column height) and inversely proportional to J₂.

For a tall column or a long jib, this component can be as large as or larger than the beam bending component. A common design mistake is selecting the I-beam section carefully for deflection while ignoring the column stiffness, only to find that the column flexure dominates the total result.

To reduce this component:

• Increase column pipe wall thickness (increases J₂)
• Increase column outer diameter D (most effective — J₂ ∝ D⁴)
• Reduce column height L (if installation allows)
• Reduce jib reach a

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What Excessive or Unexpected Deflection Indicates

The calculated δ represents the expected deflection of a correctly designed and assembled crane under its rated load. If the actual observed deflection during operation or inspection differs significantly from the calculated value, this discrepancy is diagnostically important:

Actual deflection > calculated: The crane may be overloaded beyond its rated capacity; bolted connections at the column flange or bracket may be loose or failing; the column base anchor bolts may have loosened or the foundation may have settled; the I-beam or column may have been substituted with a lighter section than specified; weld failures at the jib-to-column connection may have reduced structural continuity.

Actual deflection < calculated: The jib may be stiffened by unintended structural engagement with building elements (touching the wall, contact with overhead structure); the actual section sizes may be larger than documented; the load at the time of measurement may have been less than rated capacity.

Intermittent or increasing deflection over time: Progressive connection loosening, fatigue crack growth at weld toes, or foundation deterioration. Requires immediate structural inspection.

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Worked Example

Given:

• Safety factor k = 1.15
• Rated capacity Q = 5,000 N (approx. 500 kg)
• Hoist weight Gh = 800 N
• Jib radius a = 3,000 mm
• Column height L = 4,000 mm
• E = 206,000 N/mm²
• I-beam: 25b → J₁ = 5,280 cm⁴ = 52,800,000 mm⁴
• Column pipe: D = 219 mm, d = 199 mm

Step 1 — Design load p: p = 1.15 × 5,000 + 800 = 5,750 + 800 = 6,550 N

Step 2 — Column inertia J₂: J₂ = π/64 × (219⁴ − 199⁴) = π/64 × (2,289,726,321 − 1,568,239,201) = π/64 × 721,487,120 ≈ 35,440,000 mm⁴

Step 3 — Beam bending component: pa³ / (3EJ₁) = 6,550 × 27,000,000,000 / (3 × 206,000 × 52,800,000) = 176,850,000,000,000 / 32,630,400,000,000 ≈ 5.42 mm

Step 4 — Column flexure component: pa²L / (EJ₂) = 6,550 × 9,000,000 × 4,000 / (206,000 × 35,440,000) = 235,800,000,000,000 / 7,300,640,000,000 ≈ 32.30 mm

In this example the column flexure component dominates. Total δ ≈ 37.7 mm.

Step 5 — Check against limit: Allowable = a / 350 = 3,000 / 350 = 8.57 mm → Does not comply.

The column is undersized for this combination of jib radius and column height. Increasing the column outer diameter from 219 mm to 273 mm (a common next standard pipe size) increases J₂ by approximately 2.5×, reducing the column flexure component to about 13 mm and bringing the total closer to the limit. Further increase or reducing column height is needed to fully comply.

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الأسئلة الشائعة

Q1：Why does jib reach (a) affect deflection so much more than column height (L)?

The beam bending component is proportional to a³, while the column flexure component is proportional to a²L. For most practical jib cranes, a and L are of similar magnitude, so reach dominates through the cubic term. Doubling the jib reach from 2 m to 4 m increases beam bending deflection by 8× and column-induced deflection by 4×. This is why the allowable deflection limit is expressed as a fraction of jib reach (a/350), not as an absolute value.

Q2：Can this calculator be used for wall-mounted jib cranes?

The formula applies directly to free-standing pillar jib cranes where the column is fixed at the base and carries the full bending moment. Wall-mounted jib cranes are structurally different: the jib arm is supported by two bracket connections to the wall, which transfer the load as a force couple rather than a bending moment. The column flexure term does not apply. For wall-mounted jibs, only the cantilever beam bending component (pa³ / 3EJ₁) is relevant, and the wall bracket reactions and wall structure capacity must be checked separately.

Q3：What elastic modulus should I use for the column and beam?

For all standard structural steels — Q235, Q345, S235, S355, A36, A572 Gr.50 — the elastic modulus E is 206,000 N/mm² (206 GPa). This value is essentially identical across the full range of structural steel grades because E is a property of the iron crystal lattice, not the alloy composition. Only when using stainless steel (E ≈ 193 GPa), aluminium alloy (E ≈ 70 GPa), or other non-ferrous structural materials does E differ meaningfully.

Q4：My calculated deflection exceeds the limit — what are my options?

Identify which component dominates. If beam bending dominates: select the next larger I-beam section (J₁ increases significantly with each step up). If column flexure dominates: increase the column pipe outer diameter D (most effective, since J₂ ∝ D⁴), increase wall thickness, or reduce column height if the installation allows. Reducing jib reach a is the most structurally efficient option but may not be operationally acceptable. Verify the inputs — a common error is entering a in metres instead of millimetres, which understates deflection by a factor of 10⁶.

Q5：How do I find the effective slewing radius a?

a is the horizontal distance from the column centreline to the hook centreline at maximum trolley outreach. It is not the jib beam length — the beam length is slightly longer because the beam extends past the column centreline on both sides (or because the hoist cannot reach exactly to the beam end). The manufacturer's dimensional drawing will show the "effective slewing radius" or "working radius" explicitly. If working from first principles, a = (beam length / 2) − (column radius) − (minimum hoist approach from beam end).

Q6：The result differs from my FEM analysis — which is correct?

The formula used here is a two-component analytical model based on linear elastic beam theory. It assumes the column is fully fixed at the base (no foundation rotation), the jib-to-column connection is rigid (fully continuous moment transfer), and both members behave as prismatic beams. A finite element model (FEM) that correctly captures base fixity, connection flexibility, and the 3D interaction between beam and column will generally give a more accurate result. Discrepancies of 10–20% are normal; larger discrepancies suggest that the analytical model's assumptions (base fixity, rigid connection) do not match the actual structure and should be investigated.