Mechanical Engineering Design · Part 3 — Design of Mechanical Elements

Welding, Bonding, and the Design of Permanent Joints

Analysis and design of welded connections: the throat-shear model for butt and fillet welds, the treat-the-weld-as-a-line method for joints in torsion and bending, electrode and code-allowable strengths, static and fatigue checks, plus resistance welding and adhesive-bonded lap joints.

9–1Welding Symbols

Learning objectives

Translate a joint drawing into weld type, side, size, effective length, and inspection intent.
Construct a weld-group free-body diagram and separate direct force from eccentric moment.
Calculate nominal throat stress and distinguish it from local toe or root stress.
Select static or fatigue resistance data with traceable code, procedure, or test provenance.
Iterate geometry and verify weld metal, base metal, stiffness, access, and reliability constraints.

Symbols are manufacturing requirements

A welding symbol identifies joint form, arrow side or other side, size, effective length, contour, all-around or field status, and a tail reference. It is not merely analysis notation: ambiguity changes heat input, access, inspection, and the load path. The design calculation and drawing must use the same effective geometry.

Physical meaning

A permanent joint redirects force through deposited metal, a heat-affected zone, or a cured bond line. The symbol defines what should exist; procedure qualification and inspection establish what was actually produced.

9–2Butt and Fillet Welds

Effective resisting area

A full-penetration groove weld is commonly modeled over effective throat thickness \(t\) and length \(l\), giving average normal or shear stress. For an equal-leg 45-degree fillet of leg \(h\), the shortest throat is \(t=0.707h\). Reinforcement and leg area are not substituted for qualified effective throat.

Nominal throat stress

\[\sigma_{\mathrm{nom}}=\frac{F_n}{tl},\qquad \tau_{\mathrm{nom}}=\frac{F_s}{tl},\qquad t=0.707h\ \text{(equal-leg fillet)}\]

Assumptions and limits

Average throat stress assumes a defined effective length, compatible deformation, and a load path represented by resultants. It does not resolve toe and root singularities, residual stress, porosity, undercut, lack of fusion, or heat-affected-zone property changes. Those are addressed through detail classification, code provisions, qualified procedures, inspection, and fatigue data.

FIG. 9.1Fillet-weld geometry: leg size h, with the 45° throat of width 0.707h carrying the assumed shear load. (Original diagram, redrawn.)

EXAMPLE 1Direct shear in a paired fillet weld

A fixture transfers a static vertical load of \(48\ \text{kN}\) through two parallel fillet welds, each \(80\ \text{mm}\) long with a \(5\ \text{mm}\) leg. Determine nominal throat shear and margin against a supplied project allowable of \(126\ \text{MPa}\).

Recognition strategy

The load passes through the weld-group centroid, so this is direct shear rather than eccentric weld-group torsion. The requested stress is nominal throat stress; the supplied allowable is strength information and must not be confused with the calculated stress.

Given information

Load \(F=48\ \text{kN}\), shared by two continuous weld segments.
Leg size \(h=5\ \text{mm}\), segment length \(l=80\ \text{mm}\), and 45-degree fillet throat \(t=0.707h\).
Project allowable \(\tau_{\mathrm{allow}}=126\ \text{MPa}\), assumed applicable to this electrode, process, load direction, and static service.

Model and interpretation

Isolate the attached plate. Its free-body diagram has downward applied load and an equal upward resultant distributed over both effective throats. Uniform average shear is assumed; starts, stops, root defects, and local toe stresses are outside this nominal model.

Step-by-step solution

1. Compute the effective throat width from the specified leg: \(t=0.707(5\ \text{mm})=3.535\ \text{mm}\). The leg itself is not the resisting thickness.

2. Sum both effective lengths before forming area: \(L=2(80)=160\ \text{mm}\), hence \(A_t=tL=3.535(160)=565.6\ \text{mm}^2\).

3. Apply equilibrium and the direct-shear model: \(\tau=F/A_t=(48{,}000\ \text{N})/(565.6\ \text{mm}^2)=84.87\ \text{MPa}\).

4. Compare demand with supplied resistance: utilization is \(84.87/126=0.674\), and the allowable-to-demand margin is \(126/84.87=1.48\).

Verification

Units: Because \(1\ \text{N}/\text{mm}^2=1\ \text{MPa}\), dividing newtons by square millimetres produces the reported stress without a hidden conversion.

Signs and direction: The throat traction acts upward on the loaded plate and balances the downward load. The reported positive value is its magnitude, not a tensile normal stress.

Plausibility: Each weld carries \(24\ \text{kN}\); dividing that by one throat area \(282.8\ \text{mm}^2\) independently reproduces \(84.87\ \text{MPa}\).

Final result: \(\boxed{\tau_{\mathrm{nom}}=84.9\ \text{MPa},\quad \tau_{\mathrm{allow}}/\tau_{\mathrm{nom}}=1.48}\).

Engineering interpretation

The weld passes this one static nominal-shear comparison, but approval still requires the governing welding code, minimum-size rules, effective-length rules, base-metal checks, qualified procedure, and inspection acceptance criteria.

9–3Stresses in Welded Joints in Torsion

Weld-group mechanics

Move the external force to the weld-group centroid. The equivalent system is the same force plus moment \(M=Fe\). Direct shear is uniform in the elastic line model; moment-induced shear is tangent to the radius and proportional to distance from the centroid.

Direct and moment-induced throat shear

\[\tau'=\frac{V}{A_t},\qquad \boldsymbol{\tau}''=\frac{M}{J}( -y\,\mathbf{i}+x\,\mathbf{j}),\qquad J=0.707hJ_u\]

The sign of each vector component matters. Evaluate candidate points at weld ends and farthest radii, then compare resultant magnitudes rather than automatically adding scalar maxima.

EXAMPLE 2Eccentric load on a two-line weld group

A wall bracket uses two horizontal \(6\ \text{mm}\) fillet welds, each \(100\ \text{mm}\) long and separated vertically by \(60\ \text{mm}\). A downward \(20\ \text{kN}\) load acts \(120\ \text{mm}\) to the right of the weld-group centroid. Find the largest resultant nominal throat shear.

Recognition strategy

The offset load supplies both a force and a moment at the weld-group centroid. Therefore primary direct shear must be vector-added to secondary torsional shear; adding magnitudes everywhere would lose the direction information.

Given information

Two line segments span \(-50\le x\le50\ \text{mm}\) at \(y=\pm30\ \text{mm}\).
Fillet leg \(h=6\ \text{mm}\), load \(V=20\ \text{kN}\), and eccentricity \(e=120\ \text{mm}\).
Coordinates are \(+x\) right and \(+y\) upward; the applied force is downward.

Model and interpretation

Replace each 45-degree fillet by a throat strip of width \(0.707h\). The symmetric group centroid is the origin. Primary shear is uniform and downward; the clockwise moment generates tangent secondary shear proportional to radius.

Step-by-step solution

1. The effective area is \(A=0.707h(2L)=0.707(6)(200)=848.4\ \text{mm}^2\), giving primary shear \(\tau'=20{,}000/848.4=23.57\ \text{MPa}\) downward.

2. For the two unit-width lines, \(J_u=2[L^3/12+(30\ \text{mm})^2L]=346{,}667\ \text{mm}^3\), so \(J=0.707hJ_u=1.4704\times10^6\ \text{mm}^4\).

3. The centroidal moment magnitude is \(M=Ve=(20{,}000)(120)=2.40\times10^6\ \text{N}\!\cdot\!\text{mm}\). At \((50,30)\ \text{mm}\), secondary components have magnitudes \(My/J=48.97\ \text{MPa}\) and \(Mx/J=81.61\ \text{MPa}\).

4. At the endpoint where the vertical secondary component points downward, combine components: \(\tau_{\max}=\sqrt{48.97^2+(81.61+23.57)^2}=116.0\ \text{MPa}\).

Verification

Units: The line property \(J_u\) has units \(\text{mm}^3\); multiplying by throat width gives \(\text{mm}^4\), so \(Mr/J\) reduces to \(\text{N}/\text{mm}^2\).

Signs and direction: A clockwise applied moment produces tangent weld traction. At one right endpoint its vertical component reinforces downward primary shear; at the opposite endpoint it partly cancels.

Plausibility: The corner radius is \(58.3\ \text{mm}\), giving secondary magnitude \(Mr/J=95.2\ \text{MPa}\); the resolved components \(49.0\) and \(81.6\ \text{MPa}\) reproduce that magnitude.

Final result: \(\boxed{\tau_{\max}=116\ \text{MPa}}\), at the endpoint where torsional and direct vertical shear reinforce.

Engineering interpretation

Eccentricity, not direct load alone, dominates this joint. Increasing vertical separation raises the group polar moment efficiently, but any redesign must also consider bracket stiffness, weld access, termination quality, base-metal tear-out, and fatigue at the weld toe.

9–4Stresses in Welded Joints in Bending

Line-group bending model

For an in-plane weld pattern resisting bending, compute its centroid and unit-width second moment \(I_u\), then scale by throat width: \(I=tI_u\). The nominal bending traction varies linearly with distance \(c\), while direct shear remains \(V/A_t\).

Bending distribution

\[\tau_b=\frac{Mc}{I},\qquad I=0.707hI_u,\qquad \boldsymbol{\tau}_{\mathrm{res}}=\boldsymbol{\tau}'+\boldsymbol{\tau}_b\]

This elastic distribution is suitable for screening a sufficiently stiff connection. Flexible plates can redistribute load and introduce prying; finite-element refinement is useful only when mesh, contact, weld representation, and acceptance criteria are documented.

9–5The Strength of Welded Joints

Demand is not resistance

Calculated \(\sigma\) or \(\tau\) is load-induced stress. Electrode classification, base-metal yield or ultimate strength, and code allowable are resistance data. Never infer an allowable from electrode name alone: edition, process, position, filler, base-metal compatibility, temperature, and procedure qualification control applicability.

Generic allowable comparison

\[U_{\mathrm{weld}}=\frac{\tau_{\mathrm{demand}}}{\tau_{\mathrm{allow,weld}}}\le1,\qquad U_{\mathrm{base}}=\frac{\sigma_{\mathrm{demand}}}{\sigma_{\mathrm{allow,base}}}\le1\]

Data provenance

Code coefficients, fatigue categories, minimum weld sizes, and electrode properties must be cited from the governing edition or approved project specification. Educational values supplied in an example apply only to that example and are not reusable design constants.

9–6Static Loading

Problem-solving strategy

Draw the connected part as a free body and reduce loads to force and moment at the weld-group centroid.
Define effective throats and lengths from the intended symbol, excluding unqualified reinforcement.
Calculate direct and secondary vector stresses at all credible critical locations.
Compare weld demand with traceable resistance, then check base metal, block shear, bearing, and tear-out.
Iterate leg, length, pattern, and plate stiffness; finally audit units, access, distortion, and inspection.

A static pass is not a fatigue pass. Local yielding may redistribute a ductile weld group under a monotonic load, but brittle fracture, lamellar tearing, and low-temperature toughness can prevent that beneficial redistribution.

9–7Fatigue Loading

Detail category governs fatigue

Cyclic weld behavior is controlled by stress range, geometry, toe and root quality, residual stress, mean-stress treatment embedded in the adopted method, and inspected defect population. Nominal static throat stress alone does not predict life. Use the applicable code detail category or qualified test curve and preserve its definitions of stress range and cycle count.

Do not combine a local notch factor with a detail-category curve that already includes weld geometry unless the governing method explicitly requires it; that double-counts concentration. Screen both the weld and adjacent base-metal detail.

9–8Resistance Welding

Nugget load paths

Spot and seam welding create fused nuggets by controlled current, pressure, and time. A spot pattern should transfer load primarily by nugget shear rather than peel. Nugget diameter, pitch, edge distance, sheet thickness, electrode wear, shunting, and destructive qualification determine resistance; vendor or process data must define per-spot capacity.

9–9Adhesive Bonding

Shear lag, peel, and durability

Adhesives distribute load and join dissimilar materials, but lap-joint shear is highest near overlap ends because adherends stretch. Longer overlap therefore gives diminishing strength gains. Symmetric double laps, tapers, and spew fillets reduce eccentric peel, which is often more damaging than average shear.

Qualified peak-factor screening model

\[\tau_{\mathrm{avg}}=\frac{P}{bL},\qquad \tau_{\max}=\beta\tau_{\mathrm{avg}}\]

The factor \(\beta\) is not universal; it must come from a validated analysis or tests matching adhesive, adherends, thickness, cure, environment, and geometry.

EXAMPLE 3Selecting overlap for a qualified adhesive joint

A bonded sensor bracket must transmit \(9.0\ \text{kN}\) in shear through a symmetric double-lap joint having two \(35\ \text{mm}\)-wide bond lines. Qualification tests for the exact adhesive, adherend, cure, temperature, and surface process provide a peak allowable shear of \(12\ \text{MPa}\) and a validated peak-to-average factor \(\beta=2.40\). Find the minimum overlap and audit a manufacturable \(30\ \text{mm}\) choice.

Recognition strategy

The qualification factor signals nonuniform shear-lag rather than uniform average stress. Symmetry makes each of the two interfaces carry half the total load, so use \(\tau_{\max}=\beta P/(2bL)\) while recognizing that peel and environmental durability remain separate failure modes.

Given information

Transferred load \(P=9{,}000\ \text{N}\) and two bond lines, each of width \(b=35\ \text{mm}\).
Qualified peak allowable \(12\ \text{MPa}\) and dimensionless factor \(\beta=2.40\); both are supplied test data, not invented handbook constants.
Candidate overlap \(L=30\ \text{mm}\), with symmetric loading assumed to split load equally and suppress gross peel.

Model and interpretation

Isolate the central adherend and represent each interface by distributed shear whose resultant is \(P/2\). The validated factor maps average interface stress to the governing end peak for this qualified configuration.

Step-by-step solution

1. Impose the peak criterion symbolically: \(\beta P/(2bL)\le\tau_{\mathrm{allow}}\), so \(L_{\min}=\beta P/(2b\tau_{\mathrm{allow}})\).

2. Substitute the qualified data: \(L_{\min}=2.40(9{,}000)/[2(35)(12)]=25.71\ \text{mm}\). This is an analytical minimum, not yet a drawing dimension.

3. For the selected \(30\ \text{mm}\) overlap, \(\tau_{\mathrm{avg}}=9{,}000/[2(35)(30)]=4.286\ \text{MPa}\) and \(\tau_{\max}=2.40(4.286)=10.29\ \text{MPa}\).

4. The peak utilization is \(10.29/12=0.857\), leaving \(14.3\%\) stress margin. Record the explicit decision: accept \(30\ \text{mm}\) only if process and peel requirements also pass.

Verification

Units: The total bonded area \(2bL\) is in square millimetres, so load divided by that area is \(\text{N}/\text{mm}^2=\text{MPa}\); the qualification multiplier carries no units.

Signs and direction: Reversing the load reverses bond shear but not the calculated magnitude. A double-lap arrangement is specified so the primary load path has negligible eccentric peel.

Plausibility: The selected overlap exceeds the analytical \(25.71\ \text{mm}\) minimum, and its computed peak \(10.29\ \text{MPa}\) is correspondingly below the \(12\ \text{MPa}\) limit.

Final result: \(\boxed{L_{\min}=25.7\ \text{mm};\ \text{select }L=30\ \text{mm},\ \tau_{\max}=10.3\ \text{MPa}}\).

Engineering interpretation

The modest numerical margin makes process control decisive. Drawing release should specify surface preparation, bond-line thickness, cure verification, edge geometry, inspection, and traceable qualification over moisture and temperature; the simplified shear model cannot certify peel or long-term durability.

Chapter recap

Reduce eccentric loading at the weld-group centroid and combine throat stresses vectorially.
Keep nominal demand distinct from local stress and from code- or test-based resistance.
Iterate geometry together with base-metal, fatigue, access, distortion, and quality constraints.

Before moving on

You should be able to read the joint definition, draw its load path, select the correct throat or bond model, compute and audit demand, cite resistance provenance, and state what inspection or qualification remains.