8.1 Welded Joints & Weld Group Stress Analysis

8.1 Welded Joints & Weld Group Stress Analysis

Quick Summary: In structural machine design, welded connections provide permanent, high-integrity joints capable of transmitting static and dynamic forces across structural members. Fillet welds represent the most prevalent permanent fastening configuration, engineered under the foundational principle that fillet welds fail exclusively in shear across their minimum theoretical throat dimension ($t_e = 0.707 h$). This section establishes the mechanical fundamentals of fusion welding, groove and fillet geometry, AWS electrode classifications, code allowables, and the "treated as lines" method for analyzing weld groups subjected to in-plane and out-of-plane eccentric load vectors.

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1. Fusion Welding Processes: Mechanisms, Capabilities & Metallurgy

Fusion welding coalesces two or more workpieces by localized melting of base metals, typically augmented with molten filler metal. The four primary industrial electric arc welding processes used in mechanical and structural fabrication include:

Shielded Metal Arc Welding (SMAW / "Stick")

• Operating Mechanism: An electric arc strikes between a flux-covered consumable metal electrode and the workpiece pool. The intense arc temperature ($> 3000^\circ\text{C}$) melts the core wire and base metal while vaporizing the chemical flux coating. The flux decomposition generates a protective gaseous envelope ($CO_2$, $H_2O$) and a liquid slag blanket that prevents atmospheric oxygen and nitrogen contamination of the molten puddle.
• Strengths & Limitations: Highly portable, inexpensive equipment, robust in outdoor field environments. However, deposition rates are low, duty cycles are constrained by periodic rod changes, and slag must be mechanically chipped between successive passes, introducing potential slag inclusion defects.

Gas Metal Arc Welding (GMAW / "MIG")

• Operating Mechanism: A continuous, mechanically driven solid bare wire electrode feeds through a welding gun coaxially with an externally supplied shielding gas (argon, helium, $CO_2$, or balanced mixtures like $75\%\text{ Ar} / 25\%\text{ CO}_2$).
• Transfer Modes: Short-circuit transfer (low heat input, thin sheet metals), globular transfer (erratic, high spatter), and spray transfer (high current, argon-rich gas, smooth droplet transfer, deep joint penetration).
• Strengths & Limitations: High deposition rate, continuous operation without slag chipping, high operator efficiency, easily automated by robotics. Sensitive to wind drafts that disperse the external shielding gas, causing severe porosity.

Gas Tungsten Arc Welding (GTAW / "TIG")

• Operating Mechanism: An arc is sustained between a non-consumable tungsten or tungsten-alloy electrode ($W\text{-}ThO_2$, $W\text{-}La_2O_3$) and the weld puddle beneath a continuous stream of inert gas (pure argon or helium). Filler metal is introduced externally by hand or mechanical wire feeder.
• Strengths & Limitations: Delivers the highest metallurgical quality, superior cosmetic finish, precise heat input control, and zero spatter or slag. Ideal for root passes in critical high-pressure piping, thin-gauge sheet, reactive metals (titanium, zirconium), and aluminum alloys. Disadvantages include very low deposition rates, high equipment and labor costs, and high operator skill requirements.

Flux-Cored Arc Welding (FCAW)

• Operating Mechanism: Combines continuous wire feeding with flux protection. A hollow, tubular consumable wire contains core flux agents, deoxidizers, and slag-forming compounds. Can operate self-shielded (FCAW-S) or gas-shielded (FCAW-G, "dual shield").
• Strengths & Limitations: Exceptionally high deposition rates (often exceeding $10\text{--}15\text{ lb/hr}$), deep root penetration, excellent tolerance of surface rust and mill scale, outstanding for heavy structural steel fabrication and field erection.

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2. Weld Joint Geometry & Penetration Classifications

Welded joints are classified geometrically by how the mating workpieces are oriented relative to one another:

Butt Joint (Groove Weld)        Lap Joint (Fillet Welds)        Tee Joint (Fillet Welds)
+------+        +------+        +-----------+                  +-------+
| Base | [Weld] | Base |        |  Plate 1  | [Fillet]         | Plate | (Stem)
| Matl |========| Matl |        +-----------+====+             +---+---+
+------+        +------+              +----------+                 | | [Fillets]
                                      | Plate 2  |            =====+=+=====
                                      +----------+            +-----------+
                                                              | Plate (Flange)

Groove (Butt) Welds

Groove welds join plates edge-to-edge. Edge preparations include square grooves (for thin plates $< 6\text{ mm}$), single-V and double-V grooves, single-bevel and double-bevel, U-grooves, and J-grooves for thick plates to ensure complete heat and filler penetration.

• Complete Joint Penetration (CJP): The weld metal extends through the entire thickness of the joined plates, completely fusing across the cross-section. Backing bars or root back-gouging are frequently specified to ensure sound root fusion. In static tension or compression, a CJP groove weld fabricated with matching filler metal has an allowable strength equal to or exceeding that of the weaker base metal; thus, its joint efficiency is taken as $100\%$ ($F_w = F_{\text{base}}$).
• Partial Joint Penetration (PJP): The weld penetration is shallower than the full material thickness, leaving an unpenetrated root gap. PJP groove welds are economical for thick members where applied stresses are moderate. However, the unpenetrated root acts like a built-in crack, a severe notch under tension normal to the weld. PJP welds are therefore avoided where cyclic tension acts perpendicular to the joint axis, and AWS D1.1 restricts them in cyclically loaded connections.

Fillet Welds

Fillet welds join surfaces positioned approximately at right angles ($90^\circ$), such as in lap joints, tee joints, and corner joints. Fillet welds are the workhorse of mechanical machinery because they require no edge preparation (beveling or chamfering), significantly reducing fabrication machining costs.

Plug and Slot Welds

Used to join overlapping plates by filling a circular hole (plug) or elongated slot (slot) machined into one member with weld metal. Plug welds are designed strictly for direct shear transfer parallel to the faying surface; they cannot be relied upon to resist tensile prying forces normal to the joint plane.

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3. Fillet Weld Mechanics & The Throat Shear Failure Criterion

Fillet Weld Cross-Sectional Geometry

A standard equal-leg fillet weld exhibits a right-triangular cross section:

                 |
                 | \ 
          Leg    |   \ 
          Size   |  t_e\  Weld Face (Convex/Concave/Flat)
           h     |       \ 
                 |         \ 
                 +-----------+---
                      Leg Size h

• Leg Size ($h$ or $w$): The length of the weld legs along the two perpendicular fusion faces.
• Theoretical Throat ($t_e$): The perpendicular distance from the joint root (intersection of the fusion faces) to the hypotenuse of the largest inscribed right triangle:

$$t_e = h \cos(45^\circ) = \frac{h}{\sqrt{2}} \approx 0.7071 h$$

• Effective Throat vs. Actual Throat: The actual throat includes weld face convexity (external reinforcement). However, codes stipulate that convexity does not contribute to design strength. Excess convexity creates sharp notch re-entrant corners at the weld toes, accelerating fatigue crack initiation. In concave welds, the effective throat is measured from the root to the lowest point of the concave face.
• Effective Length ($L$): The total length of the full-size weld segment. To account for start and stop craters where the weld cross section is undersized, some design texts deduct one leg size at each end ($L_{\text{eff}} \approx L_{\text{nominal}} - 2h$). AISC 360 instead counts the full length, including end returns, but reduces the strength of very long end-loaded fillet welds (longer than $100$ times the weld size). Follow whichever rule the problem's code specifies.
• Effective Throat Area ($A_w$): The minimum load-bearing cross section:

$$A_w = t_e L = 0.7071 h L$$

The Throat Shear Failure Criterion

Consider a fillet weld subjected to external loading. Whether the applied force acts:

1. Parallel to the weld axis (transverse lap shear / parallel fillet),
2. Perpendicular to the weld axis (transverse fillet in direct tension or compression), or
3. Under combined bending and twisting moments,

Design codes simplify all of these cases by treating the weld as failing in shear across its minimum throat plane ($45^\circ$ for an equal-leg fillet), whatever the load direction.

Tests show the real behavior is more complex: transversely loaded fillets fracture on planes other than the $45^\circ$ throat and are roughly $1.5$ times stronger than longitudinal fillets. AISC 360 (Section J2) permits a directional strength increase of $(1.0 + 0.50 \sin^{1.5}\theta)$ for in-plane loads on linear weld groups, but the basic throat-shear method ignores it and stays conservative. In the basic method, all fillet weld stresses are treated as an equivalent direct shear stress acting over the theoretical throat area:

$$\tau = \frac{F}{A_w} = \frac{F}{0.7071 h L}$$

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4. AWS Electrode Classification & Code Allowable Stresses

Welding electrodes are specified using the American Welding Society (AWS) classification system. For covered SMAW electrodes:

                                    E 70 1 8
                                    - -- - -
  Electrode ------------------------+  | | |
  Minimum Tensile Strength (ksi) ------+ | |
  Welding Position Capability -----------+ |
  Coating Type, Current & Polarity --------+

• Prefix "E": Designates a welding electrode.
• First Two Digits (e.g., "70"): Minimum tensile strength $F_{EXX}$ of the deposited weld metal in kilopounds per square inch (ksi). E70XX denotes $F_{EXX} = 70\text{ ksi}$ ($483\text{ MPa}$); E60XX denotes $F_{EXX} = 60\text{ ksi}$ ($414\text{ MPa}$).
• Third Digit (e.g., "1"): Usable welding position: 1 indicates all positions (flat, horizontal, vertical, overhead); 2 indicates flat and horizontal fillet welds only; 4 indicates vertical-down.
• Fourth Digit (e.g., "8"): Chemical coating composition, arc penetration, and electrical characteristics (8 denotes low-hydrogen iron powder coating operating on AC or DCEP).

Allowable Stress Design (ASD) Code Provisions

Under the AISC / AWS Allowable Stress Design specification, structural machine joints must satisfy two separate strength checks:

1. Weld Metal Shear Capacity: The allowable shear stress on the effective throat of a fillet weld is:

$$F_w = 0.30 F_{EXX}$$

For standard electrode grades:

• E70XX: $F_w = 0.30(70\text{ ksi}) = 21.0\text{ ksi} \approx 144.8\text{ MPa}$
• E60XX: $F_w = 0.30(60\text{ ksi}) = 18.0\text{ ksi} \approx 124.1\text{ MPa}$

1. Base Metal Shear Capacity: The shear stress in the adjoining base metal along the fusion boundary must not exceed $40\%$ of the base metal yield strength:

$$F_{v,\text{base}} = 0.40 F_y$$

For common structural steels:

• ASTM A36 Steel ($F_y = 36\text{ ksi}$): $F_{v,\text{base}} = 0.40(36\text{ ksi}) = 14.4\text{ ksi}$
• ASTM A572 Grade 50 Steel ($F_y = 50\text{ ksi}$): $F_{v,\text{base}} = 0.40(50\text{ ksi}) = 20.0\text{ ksi}$

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5. Weld Groups Under Eccentric Loading: Treated as Lines Method

When a structural connection transfers loads through an arrangement of welds that does not align with the load line of action, the weld group experiences combined direct shear and secondary twisting (in-plane torsion) or tipping (out-of-plane bending).

The Treated-as-Lines Method Concept

Because fillet weld leg sizes are small relative to overall joint dimensions, the weld throat is treated as a one-dimensional line element with unit thickness ($t = 1$). Geometric properties are calculated on this unit line and subsequently scaled by the actual throat thickness $t_e = 0.707 h$:

$$A_w = 0.707 h A_u, \quad I_w = 0.707 h I_u, \quad J_w = 0.707 h J_u$$

where $A_u$, $I_u$, and $J_u$ are the unit length, unit second moment of area, and unit polar moment of inertia, respectively.

Step-by-Step Analytical Formulation

Step 1: Centroid of the Weld Group

Locate the centroid $(\bar{x}, \bar{y})$ of the weld group relative to a convenient reference origin:

$$A_u = \sum L_i$$ $$\bar{x} = \frac{\sum L_i x_i}{A_u}, \quad \bar{y} = \frac{\sum L_i y_i}{A_u}$$

where $L_i$ is the length of individual straight weld segments and $(x_i, y_i)$ are their respective segment midpoints.

Step 2: Unit Polar Moment of Inertia ($J_u$)

By the perpendicular axis theorem, the unit polar moment of inertia about the centroid is:

$$J_u = I_{ux} + I_{uy}$$

Using the parallel axis theorem for each line segment:

• For a horizontal weld segment of length $b$ positioned at vertical distance $y_i$ from the group centroid: $I_{ux,i} = b y_i^2, \quad I_{uy,i} = \frac{b^3}{12} + b (x_i - \bar{x})^2$
• For a vertical weld segment of length $d$ positioned at horizontal distance $x_i$ from the group centroid: $I_{ux,i} = \frac{d^3}{12} + d (y_i - \bar{y})^2, \quad I_{uy,i} = d x_i^2$

Summing all segments yields $I_{ux}$, $I_{uy}$, and $J_u = I_{ux} + I_{uy}$.

Step 3: Primary Direct Shear Stress ($\boldsymbol{\tau}'$)

The resultant shear force $\mathbf{V}$ is assumed to distribute uniformly across the entire effective throat area, directed parallel to $\mathbf{V}$:

$$\tau' = \frac{V}{A_w} = \frac{V}{0.7071 h A_u}$$

In rectangular components (for a downward vertical load $V$ acting along $-y$):

$$\tau'_x = 0, \quad \tau'_y = -\frac{V}{A_w}$$

Step 4: Secondary Torsional Shear Stress ($\boldsymbol{\tau}''$)

The eccentricity $e$ (perpendicular distance from the line of action of $V$ to the weld group centroid) generates a torsional couple $T = V \cdot e$. Torsion induces shear stress at any point in the weld located at radius vector $\mathbf{r}_i$ from the centroid:

$$\tau'' = \frac{T r_i}{J_w} = \frac{T r_i}{0.7071 h J_u}$$

The direction of $\boldsymbol{\tau}''$ is strictly perpendicular to the position vector $\mathbf{r}_i$, oriented in the sense of the twisting moment $T$. Resolving into orthogonal components:

$$\tau''_x = \frac{T (y_i - \bar{y})}{J_w}, \quad \tau''_y = -\frac{T (x_i - \bar{x})}{J_w} \quad (\text{for clockwise } T)$$

Step 5: Resultant Shear Stress and Vector Summation

At any point in the weld group, the total shear stress vector is the linear superposition of primary and secondary vectors:

$$\boldsymbol{\tau}_{\text{res}} = \boldsymbol{\tau}' + \boldsymbol{\tau}''$$ $$\tau_x = \tau'_x + \tau''_x, \quad \tau_y = \tau'_y + \tau''_y$$ $$\tau_{\text{resultant}} = \sqrt{\tau_x^2 + \tau_y^2} = \sqrt{(\tau'_x + \tau''_x)^2 + (\tau'_y + \tau''_y)^2}$$

Step 6: Critical Point Identification

The critical point in the weld group occurs where $\boldsymbol{\tau}'$ and $\boldsymbol{\tau}''$ align most constructively (same sign) and where the distance $r_i$ from the centroid is maximized. The joint is structurally adequate if:

$$\tau_{\text{resultant},\max} \le F_w = 0.30 F_{EXX}$$

Out-of-Plane Eccentricity (Bending Shear)

If the applied load $P$ acts out-of-plane, it generates an eccentric bending moment $M = P \cdot e_{\text{out}}$. The secondary stress on the weld throat due to bending is calculated analogously to beam flexure:

$$\tau''_{\text{bending}} = \frac{M c}{I_w} = \frac{M c}{0.7071 h I_u}$$

where $c$ is the distance from the neutral bending axis of the weld group to the extreme outer fiber.

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6. The Heat-Affected Zone (HAZ) & Weld Metallurgy

The Heat-Affected Zone (HAZ) is the portion of non-melted base metal immediately adjacent to the fusion boundary whose microstructure and mechanical properties undergo profound alteration due to the thermal cycle of welding.

+-------------+---------------------+-------------------+------------------+
| Fusion Zone |  Coarse-Grain HAZ   |  Fine-Grain HAZ   | Subcritical HAZ  |
| (Cast Weld  |  (Peak T > 1100°C;  | (Peak T 900-1100°C| (Peak T 600-720°C|
|  Metal)     |   Grain Growth)     |   Grain Refined)  |    Tempered)     |
+-------------+---------------------+-------------------+------------------+

Microstructural Gradient Across the HAZ

1. Coarse-Grained Zone (Adjacent to Fusion Line): Peak temperatures exceed $1100^\circ\text{C}$, causing rapid dissolution of precipitates and dramatic austenitic grain growth. Upon rapid cooling, this coarse structure transforms into coarse bainite or hard untempered martensite, exhibiting severely degraded notch toughness and high crack sensitivity.
2. Fine-Grained Zone: Peak temperatures span $900^\circ\text{C}$ to $1100^\circ\text{C}$ (complete austenitization followed by rapid recrystallization), refining grain size and creating a tough, desirable microstructural state.
3. Intercritical Zone: Temperatures reach $727^\circ\text{C}$ to $900^\circ\text{C}$ (between $A_1$ and $A_3$ transformation temperatures). Partial transformation into austenite yields microstructural banding and heterogeneous hardness.
4. Subcritical (Tempered) Zone: Temperatures remain below $A_1$ ($< 727^\circ\text{C}$). In quench-and-tempered or cold-worked alloys, prolonged exposure over-tempers the matrix, causing localized softening, loss of yield strength, and permanent reduction in load capacity.

Hydrogen-Induced Cracking (HIC / Cold Cracking)

HIC occurs several hours to days after welding in the HAZ of carbon and alloy steels when three conditions coincide:

1. A brittle, hard microstructure (e.g., untempered martensite with hardness $> 350\text{ HV}$),
2. Diffusible hydrogen present in the weld puddle (from atmospheric moisture, damp electrode flux, grease, or rust), and
3. High tensile residual stresses developed during joint constraint and shrinkage.

Carbon Equivalent ($CE$) & Cracking Susceptibility

The propensity of steel to form brittle martensite in the HAZ is quantified by the Carbon Equivalent formula (IIW standard):

$$CE = \%\text{C} + \frac{\%\text{Mn}}{6} + \frac{\%\text{Cr} + \%\text{Mo} + \%\text{V}}{5} + \frac{\%\text{Ni} + \%\text{Cu}}{15}$$

• If $CE < 0.40\%$: Steel has excellent weldability; cracking is rare without preheat.
• If $CE = 0.40\%\text{--}0.45\%$: Moderate cracking risk; preheat required for thick plates.
• If $CE > 0.45\%$: High cracking risk; strict preheating ($150^\circ\text{C}\text{--}250^\circ\text{C}$), low-hydrogen electrodes (E7018) and controlled interpass temperatures are required, and post-weld heat treatment (PWHT) may be specified.

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7. Nondestructive Examination (NDE) & Inspection Methods

Nondestructive Examination verifies joint soundness without compromising serviceability. The five fundamental NDE methods comprise:

1. Visual Testing (VT)

• Principle: Direct human inspection using calibrated tools (fillet weld gauges, undercut gauges, magnifying optics, boroscopes).
• Scope: Surface discontinuities: weld size, throat contour, excessive reinforcement, undercut, surface porosity, overlap, visible cracking.
• Limitation: Strictly limited to outer visible surfaces; cannot detect subsurface flaws.

2. Liquid Penetrant Testing (PT / Dye Penetrant)

• Principle: Capillary action draws a low-viscosity visible or fluorescent dye into surface-breaking discontinuities. After dwell time ($10\text{--}30\text{ min}$), excess surface penetrant is washed off, and a chalky developer is applied to draw entrapped dye out by reverse capillary bleed-out.
• Scope: Detects tight surface cracks, pinholes, and porosity on any non-porous material (ferrous, non-ferrous, plastics, ceramics).
• Limitation: Cannot detect subsurface flaws or flaws packed with slag, rust, or paint.

3. Magnetic Particle Testing (MT)

• Principle: A magnetic field (via yoke or prods) is induced into the joint. Discontinuities disrupt magnetic lines of flux, creating magnetic leakage fields at the surface. Finely divided ferromagnetic particles (dry powder or fluorescent wet suspension) dusted over the joint cluster at flux leaks, visually outlining flaws.
• Scope: Fast, highly sensitive detection of surface and near-subsurface ($< 2\text{ mm}$ depth) cracks, seams, and lack of fusion.
• Limitation: Limited strictly to ferromagnetic metals (carbon steel, low-alloy steel). Ineffective on austenitic stainless steel, aluminum, or copper alloys.

4. Radiographic Testing (RT / X-Ray)

• Principle: High-energy ionizing radiation (gamma-ray isotopes like $Ir\text{-}192$, $Co\text{-}60$ or X-ray tubes) penetrates the weld thickness onto photographic film or digital detector plates. Dense sound metal attenuates radiation; voids, cracks, or slag attenuate less, appearing as darker density features.
• Scope: Volumetric inspection of internal voids, porosity, slag inclusions, and lack of penetration. Produces a permanent digital or film archival record.
• Limitation: Radiation safety and barricade requirements; expensive; insensitive to planar cracks oriented perpendicular to the radiation beam.

5. Ultrasonic Testing (UT / Phased Array PAUT)

• Principle: High-frequency piezoelectric transducers ($1\text{--}10\text{ MHz}$) propagate pulsed acoustic compression or shear waves through the weld. Acoustic impedance mismatches at crack boundaries reflect sound back to the transducer, displaying flaw depth and signal amplitude.
• Scope: Exceptional sensitivity to internal planar cracks, lack of sidewall fusion, and laminations at any depth in thick sections.
• Limitation: Highly dependent on inspector skill; requires couplant medium; coarse-grained austenitic metals scatter ultrasound.

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8. Worked Engineering Scenario: Eccentrically Loaded Weld Group Analysis

               Weld #1: b = 4.0 in
       ============================== (y = +3.0 in)
       |                            |
       |                            |
       |         Centroid           |                        |
       |          (2, 0)            |                        | P = 12.0 kips
       |            *               |                        | (Downward)
       |                            |                        v
       |                            |
       ============================== (y = -3.0 in)
       |<-------- b = 4.0 in ------->|<------ 6.0 in ------->|
                                     Line of Action (x = 10.0 in)

Engineering Problem Statement

A structural mounting bracket is joined to a heavy vertical steel column by two parallel horizontal fillet welds, each of length $b = 4.0\text{ in}$, separated by a vertical spacing of $d = 6.0\text{ in}$. The welds are placed at $y = +3.0\text{ in}$ and $y = -3.0\text{ in}$, extending from $x = 0$ to $x = 4.0\text{ in}$.

The bracket supports a steady downward vertical load of $P = 12.0\text{ kips}$ ($12,000\text{ lbf}$) acting at horizontal coordinate $x = 10.0\text{ in}$ (which lies $6.0\text{ in}$ to the right of the weld toe, or an eccentricity of $e = 8.0\text{ in}$ from the weld group centroid). The specified electrode is AWS E70XX ($F_{EXX} = 70.0\text{ ksi}$), and the fillet weld leg size is $h = 5/16\text{ in} = 0.3125\text{ in}$.

Required Engineering Output:

1. Calculate the unit geometric properties ($A_u, I_{ux}, I_{uy}, J_u$) and effective throat properties ($A_w, J_w$).
2. Compute the primary direct shear stress vector $\boldsymbol{\tau}'$.
3. Compute the secondary torsional shear stress vector $\boldsymbol{\tau}''$ at all outer corners.
4. Determine the critical stress location and calculate the maximum resultant shear stress $\tau_{\text{res},\max}$.
5. Check joint code compliance against AISC/AWS allowable weld shear stress $F_w$ and evaluate the factor of safety $n$.

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Step-by-Step Solution

Step 1: Weld Group Geometry & Centroid Location

The weld group consists of two identical horizontal lines:

• Weld 1: Length $L_1 = 4.0\text{ in}$, located from $(0, 3.0)$ to $(4.0, 3.0)$. Midpoint: $(2.0, 3.0)$.
• Weld 2: Length $L_2 = 4.0\text{ in}$, located from $(0, -3.0)$ to $(4.0, -3.0)$. Midpoint: $(2.0, -3.0)$.

1. Total Unit Length ($A_u$): $A_u = L_1 + L_2 = 4.0\text{ in} + 4.0\text{ in} = 8.0\text{ in}$

2. Centroid Coordinates ($\bar{x}, \bar{y}$): $\bar{x} = \frac{L_1 x_1 + L_2 x_2}{A_u} = \frac{(4.0)(2.0) + (4.0)(2.0)}{8.0} = 2.0\text{ in}$ $\bar{y} = \frac{L_1 y_1 + L_2 y_2}{A_u} = \frac{(4.0)(+3.0) + (4.0)(-3.0)}{8.0} = 0.0\text{ in}$ The weld group centroid sits at $(\bar{x}, \bar{y}) = (2.0\text{ in}, 0.0\text{ in})$.

Step 2: Unit Moments of Inertia ($I_{ux}, I_{uy}, J_u$)

1. Unit Moment of Inertia about x-axis ($I_{ux}$): Both welds are horizontal lines parallel to the x-axis ($I_{x,\text{own}} = 0$). By the parallel axis theorem: $I_{ux} = \sum L_i y_i^2 = L_1 y_1^2 + L_2 y_2^2 = (4.0\text{ in})(+3.0\text{ in})^2 + (4.0\text{ in})(-3.0\text{ in})^2$ $I_{ux} = 4.0(9.0) + 4.0(9.0) = 36.0 + 36.0 = 72.0\text{ in}^3$

2. Unit Moment of Inertia about y-axis ($I_{uy}$): Each line segment has length $b = 4.0\text{ in}$. Since their midpoints lie directly on $\bar{x} = 2.0\text{ in}$, the parallel axis distance is zero ($x_i - \bar{x} = 0$): $I_{uy} = \sum \left( \frac{L_i^3}{12} + L_i (x_i - \bar{x})^2 \right) = 2 \times \left( \frac{4.0^3}{12} + 0 \right) = 2 \times \left(\frac{64.0}{12}\right) = 2 \times 5.3333 = 10.667\text{ in}^3$

3. Unit Polar Moment of Inertia ($J_u$): $J_u = I_{ux} + I_{uy} = 72.0\text{ in}^3 + 10.667\text{ in}^3 = 82.667\text{ in}^3$

Step 3: Effective Weld Throat Properties

For a leg size of $h = 5/16\text{ in} = 0.3125\text{ in}$:

• Theoretical throat thickness: $t_e = 0.7071 h = 0.7071 \times 0.3125\text{ in} = 0.22097\text{ in}$
• Effective weld area ($A_w$): $A_w = t_e A_u = (0.22097\text{ in})(8.0\text{ in}) = 1.7678\text{ in}^2$
• Effective polar moment of inertia ($J_w$): $J_w = t_e J_u = (0.22097\text{ in})(82.667\text{ in}^3) = 18.267\text{ in}^4$

Step 4: Primary Direct Shear Stress Vector ($\boldsymbol{\tau}'$)

The applied load $P = 12.0\text{ kips}$ acts downward ($-y$). The direct shear stress is uniform across the throat area: $\tau'_x = 0$ $\tau'_y = -\frac{P}{A_w} = -\frac{12.0\text{ kips}}{1.7678\text{ in}^2} = -6.7881\text{ ksi} \approx -6.79\text{ ksi}$

Step 5: Torsional Moment & Secondary Shear Stress ($\boldsymbol{\tau}''$)

1. Eccentricity ($e$): The load line is at $x = 10.0\text{ in}$. The centroid is at $\bar{x} = 2.0\text{ in}$: $e = x_{\text{load}} - \bar{x} = 10.0\text{ in} - 2.0\text{ in} = 8.0\text{ in}$

2. Torsional Couple ($T$): $T = P \cdot e = (12.0\text{ kips})(8.0\text{ in}) = 96.0\text{ kip}\cdot\text{in} \quad (\text{Clockwise})$

3. Evaluation of Outer Corner Points: The weld group has four corner points relative to origin $(0, 0)$:

   • Top-right corner $A$: $(4.0, +3.0) \implies x - \bar{x} = +2.0, \; y - \bar{y} = +3.0$
   • Bottom-right corner $B$: $(4.0, -3.0) \implies x - \bar{x} = +2.0, \; y - \bar{y} = -3.0$
   • Top-left corner $C$: $(0.0, +3.0) \implies x - \bar{x} = -2.0, \; y - \bar{y} = +3.0$
   • Bottom-left corner $D$: $(0.0, -3.0) \implies x - \bar{x} = -2.0, \; y - \bar{y} = -3.0$

   For a clockwise twisting moment $T = 96.0\text{ kip}\cdot\text{in}$: $\tau''_x = +\frac{T (y - \bar{y})}{J_w}, \quad \tau''_y = -\frac{T (x - \bar{x})}{J_w}$

   Evaluating corner $A$ ($x - \bar{x} = +2.0\text{ in}, y - \bar{y} = +3.0\text{ in}$): $\tau''_{x,A} = +\frac{(96.0\text{ kip}\cdot\text{in})(+3.0\text{ in})}{18.267\text{ in}^4} = +\frac{288.0}{18.267} = +15.766\text{ ksi}$ $\tau''_{y,A} = -\frac{(96.0\text{ kip}\cdot\text{in})(+2.0\text{ in})}{18.267\text{ in}^4} = -\frac{192.0}{18.267} = -10.511\text{ ksi}$

   Evaluating corner $B$ ($x - \bar{x} = +2.0\text{ in}, y - \bar{y} = -3.0\text{ in}$): $\tau''_{x,B} = +\frac{(96.0)(-3.0)}{18.267} = -15.766\text{ ksi}$ $\tau''_{y,B} = -\frac{(96.0)(+2.0)}{18.267} = -10.511\text{ ksi}$

Step 6: Critical Point Resultant Shear Stress

Notice that at both right-hand corners $A$ and $B$, $\tau''_y = -10.511\text{ ksi}$ is negative (downward), which acts in the exact same downward direction as the primary direct shear $\tau'_y = -6.788\text{ ksi}$:

$\tau_y = \tau'_y + \tau''_y = -6.7881\text{ ksi} + (-10.5108\text{ ksi}) = -17.2989\text{ ksi}$

For both corners $A$ and $B$, the horizontal component magnitude is $|\tau_x| = 15.766\text{ ksi}$. Therefore, both right-hand corners experience the identical maximum resultant shear stress:

$\tau_{\text{res},\max} = \sqrt{\tau_x^2 + \tau_y^2} = \sqrt{(15.766)^2 + (-17.299)^2}$ $\tau_{\text{res},\max} = \sqrt{248.57 + 299.25} = \sqrt{547.82} = 23.405\text{ ksi} \approx 23.41\text{ ksi}$

Step 7: Code Compliance & Factor of Safety Evaluation

1. Allowable Weld Shear Stress ($F_w$): For AWS E70XX electrode metal under AISC ASD: $F_w = 0.30 F_{EXX} = 0.30(70.0\text{ ksi}) = 21.00\text{ ksi}$

2. Code Compliance Check: $\tau_{\text{res},\max} = 23.41\text{ ksi} > F_w = 21.00\text{ ksi} \quad \implies \quad \textbf{OVERSTRESSED (Code Violation)}$ The current connection exceeds the code allowable shear stress by: $\frac{23.41 - 21.00}{21.00} \times 100\% = +11.48\%$

3. Redesign Recommendation: To satisfy the code requirement ($F_w \ge \tau_{\text{res}}$), the engineer must either:

   • Increase the weld leg size from $h = 5/16\text{ in}$ to $h = 3/8\text{ in} = 0.375\text{ in}$ (which reduces resultant shear to $\tau_{\text{res}} = 23.41 \times (5/6) = 19.51\text{ ksi} < 21.00\text{ ksi}$, yielding factor of safety $n = 21.00/19.51 = 1.08$), or
   • Add a vertical weld segment of length $6.0\text{ in}$ along the left base ($x = 0$) to form a channel C-shaped weld group, substantially increasing $A_u$ and $J_u$.