CELE Strength of Materials — Stresses in BeamsCheat Sheet
Cheat sheet for CELE Strength of Materials — Stresses in Beams. Compact, printable, and organised around the concepts Professional Regulation Commission (PRC) — Board of Civil Engineering tests most frequently in the CELE 2026. Perfect for the week before exam day.
Exam context
On the CELE 2026, the Strength of Materials subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Civil Engineering's pattern. Stresses in Beams lands at position 4th out of 8 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Strength of Materials on a typical CELE paper.
Stresses in Beams - Cheat Sheet
Your last-minute revision companion for the Stresses in Beams chapter. This quick-reference covers all critical formulas, stress types, section moduli, and common pitfalls tested in the PRC Civil Engineer Licensure Examination.
Sections
Formulas
Formula
σ = (M × y) / I
Meaning
σ = bending stress at distance y from neutral axis (NA); M = bending moment; I = second moment of inertia about NA
Watch Out
y = 0 at the NA, NOT at the top or bottom. Sign: sagging moment → top in compression (−), bottom in tension (+). Do NOT confuse y with c (c is extreme fiber distance).
When To Use
ANY problem asking for stress at a specific depth within a beam; y is measured from the NA (zero stress point)
Formula
σ_max = (M × c) / I = M / S
Meaning
σ_max = maximum bending stress at extreme fibers; c = distance from NA to extreme fiber; S = section modulus = I/c
Watch Out
Using σ = Mc/I instead of σ = M/S (forgetting to divide I by c to get S). Units: N·mm with mm⁴ gives MPa; kN·mm with mm⁴ gives (kN·mm/mm⁴) = kN/mm² = MPa.
When To Use
Design formula and any max-stress problem. Always use this for comparing beam capacities.
Formula
S = I / c
Meaning
Section modulus; I = second moment of inertia; c = distance to extreme fiber
Watch Out
S is section property (like a 'bending power rating'); higher S means stiffer in bending. Do NOT confuse S (section modulus) with s (spacing).
When To Use
Design: S_required = M_max / σ_allow. Pick a section with S ≥ S_required. Comparison tool for beam capacity per unit depth.
Common Values
Value
S = bh²/6
Symbol
S_rect
Quantity
Rectangular section depth multiplier for S
Value
S = πd³/32
Symbol
S_circle
Quantity
Circular section modulus multiplier
Value
4–12 MPa
Symbol
σ_allow (wood)
Quantity
Allowable bending stress (timber, typical)
Value
160–240 MPa
Symbol
σ_allow (steel)
Quantity
Allowable bending stress (mild steel, typical)
Section Title
Flexural (Bending) Stress
Important Facts
- Bending stress varies LINEARLY through depth; it is zero at the NA and reaches maximum magnitude at extreme fibers.
- For a RECTANGULAR section: I = bh³/12, c = h/2, so S = bh²/6.
- For a SOLID CIRCLE: I = πd⁴/64, S = πd³/32.
- Sagging moment (positive): bottom fiber in TENSION, top in COMPRESSION.
- Hogging moment (negative): bottom in COMPRESSION, top in TENSION.
- Deep, narrow sections are more efficient in bending than shallow, wide ones (S grows with h², not with b).
- Section modulus S is the key design parameter: σ_max = M / S, so S_required = M / σ_allow.
- The moment that makes a section 'fail' in bending is M = S × σ_allow.
Key Definitions
Term
Neutral Axis (NA)
Example
In a symmetric I-beam, the NA is at mid-height; in an unsymmetric T-beam, it is lower (closer to the heavier flange).
Definition
Longitudinal axis of a beam where bending stress = 0; passes through the centroid of the cross-section.
Term
Section Modulus (S)
Example
A W-shape has much higher S than a rectangular beam of equal area because depth matters more (S ∝ h² for rectangles).
Definition
Geometric property S = I/c measuring bending efficiency; higher S requires less moment to reach a given stress.
Term
Flexural Stress
Example
Sagging moment: top fiber compressed, bottom fiber in tension. Hogging moment: reverse.
Definition
Normal stress (tension or compression) induced by bending moment; varies linearly from zero at NA to maximum at extreme fibers.
Term
Extreme Fiber
Example
In a rectangular section, the top and bottom surfaces are extreme fibers, each at distance c = h/2 from the NA.
Definition
Outermost point(s) on a cross-section farthest from the neutral axis; experiences maximum bending stress.
Diagrams To Know
- Stress distribution diagram across beam depth: linear from −σ_max (top, compression) through zero at NA to +σ_max (bottom, tension) for sagging moment.
- Beam cross-section with NA marked and y-coordinates labeled.
- Moment sign convention: sagging (∪-shaped curvature) is positive; hogging (∩-shaped) is negative.
Reactions Or Equations
Note
Essential for unsymmetric sections (T-beams, channels) where you must locate NA first, then find I about it.
Equation
I_parallel-axis = I_centroid + A × d²
Conditions
When calculating I about an axis parallel to the centroidal axis; d = distance from centroid to new axis
Formulas
Formula
τ = (V × Q) / (I × b)
Meaning
τ = shear stress at a given level; V = transverse shear force; Q = first moment of area above (or below) that level, about NA; I = second moment of inertia; b = width at that level
Watch Out
Q is the first moment of the DETACHED part (above the level), not the whole section. Forget Q correctly and you'll get the wrong answer by a factor of 2–3. Also: τ at top and bottom = 0 (no area above/below).
When To Use
Find shear stress at ANY point in the cross-section. Most common on horizontal planes inside the beam.
Formula
τ_max (rectangle) = (3V) / (2A) = 1.5 × (V/A)
Meaning
Maximum shear stress occurs at the neutral axis for a rectangular section; A = bh = cross-sectional area
Watch Out
The factor is 1.5, NOT 1. Do not forget the 1.5 multiplier. Also, this is at the NA, not at the extreme fibers.
When To Use
Quick check of max shear in rectangular beams; often shear is not critical compared to flexure in long beams.
Formula
τ_max (solid circle) = (4V) / (3A)
Meaning
Maximum shear stress at NA for a solid circular cross-section; A = πd²/4
Watch Out
Factor is 4/3 ≈ 1.33, different from rectangle's 1.5. Know both factors.
When To Use
Solid circular shafts or timber columns; less common but always appears on boards.
Formula
τ ≈ V / A_web (wide-flange I-beam)
Meaning
Simplified: nearly all shear is carried by the thin web; A_web = thickness × depth of web
Watch Out
This is an APPROXIMATION. Exact calculation uses full τ = VQ/(Ib). Use only for rough checks.
When To Use
Steel W-sections and box girders; shear usually not critical because web area is small and shear stress capacity is high.
Common Values
Value
1.5 (i.e., τ_max = 1.5 × V/A)
Symbol
k_rect
Quantity
Rectangular shear stress factor
Value
4/3 ≈ 1.333 (i.e., τ_max = 1.333 × V/A)
Symbol
k_circle
Quantity
Circular shear stress factor
Value
0.8–2.0 MPa
Symbol
τ_allow (wood)
Quantity
Allowable shear stress (timber, typical)
Value
90–140 MPa
Symbol
τ_allow (steel)
Quantity
Allowable shear stress (mild steel, typical)
Section Title
Horizontal Shear Stress in Beams
Important Facts
- Shear stress is ZERO at top and bottom fibers (no area beyond those extremes) and MAXIMUM at the neutral axis.
- This is opposite to bending stress, which is MAXIMUM at the extreme fibers and ZERO at the NA.
- Shear distribution is NOT linear; it is parabolic (for rectangles) or other shapes depending on geometry.
- For a rectangular beam, shear stress at height y from the bottom is τ(y) = (3V)/(2A) × [1 − (2y/h − 1)²].
- At the NA (y = h/2): τ(h/2) = (3V)/(2A).
- Shear stress ALWAYS has a complementary pair: horizontal τ on horizontal planes, vertical τ on vertical planes.
- Beam failure by shear is rare in long beams but common in short, heavily loaded beams or near supports.
- Allowable shear stress (timber, typical) is much lower than allowable bending stress — often critical in short timber beams.
Key Definitions
Term
Shear Stress (τ)
Example
A horizontal plane inside a beam experiences vertical shear stress τ = VQ/(Ib); a vertical plane at the same location experiences the same horizontal τ by complement.
Definition
Internal stress on horizontal or vertical planes due to transverse shear force; complementary shear acts on both planes simultaneously.
Term
First Moment of Area (Q)
Example
For a rectangular section with depth h, at height h/2 from bottom (the NA), Q (above) = b(h/2) × (h/4) = bh²/8.
Definition
Sum of areas times their distances from a reference axis: Q = Σ(A × y). For a detached portion: Q = A_detached × y_centroid-of-detached
Term
Complementary Shear
Example
Shear on a horizontal plane equals shear on a vertical plane; both equal VQ/(Ib) at that location.
Definition
Shear stresses on perpendicular planes at a point are equal in magnitude (τ_horizontal = τ_vertical).
Term
Shear Flow (q)
Example
If q = 8 kN/m and a nail capacity is 400 N, spacing s = 400/8000 = 0.05 m = 50 mm.
Definition
q = VQ/I; the force per unit length carried by connectors in built-up beams; used to find nail/bolt spacing or weld size.
Diagrams To Know
- Shear stress distribution across beam depth: zero at top and bottom, parabolic peak at the neutral axis (for rectangles).
- Comparison diagram: bending stress (linear, max at extreme fibers) vs. shear stress (parabolic, max at NA).
- Free body diagram showing internal shear force V and element with vertical and horizontal shear stresses.
- Built-up beam cross-section with Q computed for area above a given level.
Reactions Or Equations
Note
For a rectangular part, Q = b × d_part × (depth to its centroid from NA). Always use the area and centroid ABOVE the level of interest.
Equation
Q = A_part × ȳ_part (first moment of a detached area)
Conditions
Area = A_part, distance from NA to centroid of that area = ȳ_part
Formulas
Formula
q = (V × Q) / I
Meaning
Shear flow (force per unit length) transmitted by connectors joining sections; V = shear force; Q = first moment of connected area; I = moment of inertia of full section
Watch Out
q is NOT the same as τ; q has units N/mm or kN/m (force per length), while τ has units MPa (stress). Relation: q = τ × b where b is the width.
When To Use
Size nails, bolts, welds, or glue in built-up beams (planks nailed together, flange-to-web bolts, etc.). q is the load the connector must carry per unit length.
Formula
s = F / q (or n = q / f for multiple connectors)
Meaning
Connector spacing s = capacity F divided by shear flow q. For n connectors per unit length, each must carry f = q/n.
Watch Out
Keep units consistent: if q is in N/mm and F is in N, then s is in mm. If q is in kN/m and F is in kN, then s is in m. Common error: mixing units.
When To Use
Determine nail or bolt spacing in built-up sections once q is calculated.
Common Values
Value
200–500 N per nail
Symbol
F_nail
Quantity
Typical common nail capacity (timber)
Value
40–150 kN per bolt (depends on diameter and grade)
Symbol
F_bolt
Quantity
Typical bolt capacity (shear, steel)
Value
0.6–1.0 kN per mm of weld length × 2 (front + back)
Symbol
F_weld
Quantity
Typical weld strength (fillet, steel)
Section Title
Built-Up Beams & Shear Flow
Important Facts
- Shear flow q = VQ/I represents the internal force (per unit length) that must be transmitted across a joint or connection.
- The connector (nail, bolt, weld) must be strong enough and spaced close enough to carry this flow without slipping or failure.
- In timber beams, the longitudinal shear is often the limiting factor for nailing; spacing is typically 50–200 mm for common nails.
- For bolted connections in steel, multiple bolts per connection reduce the load per bolt: q (total) = (# bolts) × (load per bolt).
- A seam or connection located farther from the neutral axis sees SMALLER Q and thus smaller q — less critical area for connectors.
- The connection at the NA (where Q is maximum) is most heavily loaded and requires the most frequent or strongest fasteners.
Key Definitions
Term
Shear Flow (q)
Example
If a nail joins two planks at a seam and q = 4 kN/m, a single nail with 200 N capacity means spacing s = 200 N / 4000 N/m = 0.05 m = 50 mm.
Definition
Force per unit length transmitted along a seam or connection in a built-up beam; q = VQ/I.
Term
Built-Up Beam
Example
Timber beams made from two-by-fours nailed together; steel plate girders with flange plates bolted to a web plate.
Definition
Beam fabricated from multiple pieces (planks, plates, angles) joined by nails, bolts, glue, or welds.
Diagrams To Know
- Built-up box section with flange and web, showing seams and connector pattern.
- Shear flow distribution along the depth: zero at top and bottom, peak at the seams near the NA.
- Typical nail spacing in timber I-beam (e.g., 150 mm on center along the web-flange junction).
Formulas
Formula
σ = (P / A) ± (M × c / I)
Meaning
Total stress = axial stress ± bending stress. P = axial force (compression positive), A = area, M = bending moment (often M = P × e, where e = eccentricity), c = distance to extreme fiber, I = moment of inertia
Watch Out
Sign convention: take compression as positive. Top fiber gets one sign, bottom the opposite. If e (eccentricity) is outside the kern (middle third for rectangles), tension appears. Do NOT forget the P/A term; many students compute only Mc/I.
When To Use
Short columns, offset loads, eccentric compression (e.g., columns on footings, bracket connections). Use + for one extreme fiber, − for the other.
Formula
e_kern = c / 3 (rectangular section)
Meaning
Kern (core) half-width; eccentricity within ± c/3 from centroid produces no tension anywhere.
Watch Out
This formula is for a RECTANGLE with the kern measured perpendicular to the direction of load. For other shapes (circular, diamond), the kern is different. Outside the kern → tension appears.
When To Use
Masonry, footing, and rigid section design where tensile stress is undesirable. For a rectangle of depth h, the middle-third rule means load must stay within h/6 of center.
Common Values
Value
c/3 = h/6 (where h = total depth in direction of load)
Symbol
e_kern
Quantity
Kern distance from centroid (rectangle)
Value
b/3 or h/3 (one-third of the width or height in the direction checked)
Symbol
kern width
Quantity
Middle-third rule width (rectangle)
Section Title
Combined Axial Load & Bending (Eccentric Loading)
Important Facts
- Combined stress is the SUPERPOSITION of uniform axial stress (P/A, same everywhere) and linear bending stress (Mc/I, zero at NA).
- At one extreme fiber: σ_max = P/A + Mc/I; at the opposite: σ_min = P/A − Mc/I (or vice versa, depending on load direction).
- If e = 0 (centric load), σ = P/A everywhere (uniform compression).
- If e ≠ 0, the NA shifts away from the geometric center. The section is NO LONGER symmetric in stress.
- Tension occurs if σ_min < 0, i.e., if P/A < Mc/I. This happens when e > c/3 (outside the kern).
- Masonry and concrete sections cannot carry tension; must keep load within the kern to avoid cracking.
- The neutral axis for combined loading is NOT at the centroid; it is where σ = 0, i.e., P/A + Mc/I = 0.
- Unsymmetric sections (T-beams, channels) have different top and bottom fiber stresses EVEN WITH CENTRIC LOADING because c_top ≠ c_bot.
Key Definitions
Term
Eccentricity (e)
Example
A 300 kN load applied 50 mm off-center on a 300×300 mm square post: e = 50 mm (dimensionless ratio e/c = 50/150 = 0.333).
Definition
Perpendicular distance from the geometric center (centroid) to the line of action of an applied axial force.
Term
Kern (or Core) of a Section
Example
For a 200 mm square, the kern extends 200/6 ≈ 33.3 mm from center; load within this region → pure compression; load outside → some tension.
Definition
Region around the centroid within which an axial load produces no tension in the section.
Term
Middle-Third Rule
Example
A 300 mm wide footing: load can be 50–250 mm from the left (middle 100 mm band out of 300 mm) to avoid tension.
Definition
For a rectangular section subject to eccentric compression, if the load line stays within the middle third (width/3 or height/3), no tension occurs.
Diagrams To Know
- Stress profile for eccentric compression: linear from tension (bottom) through zero (at shifted NA) to compression (top).
- Kern diagram: rectangle with middle-third region highlighted; load positions inside kern produce no tension.
- Column with eccentric load: offset load, resulting moment = P × e, and combined stress distribution.
- Footing with eccentric load: kern boundary shown; eccentric loads outside kern cause uplift on one edge.
Reactions Or Equations
Note
This moment is always present when the load is off-center; it is NOT an additional external moment but an inherent consequence of the eccentric load.
Equation
M = P × e (eccentric axial load produces a bending moment)
Conditions
P = axial force, e = eccentricity (perpendicular distance from centroid to load line)
Formulas
Formula
ȳ = Σ(A_i × y_i) / Σ A_i (locate NA)
Meaning
Centroid (neutral axis) location; sum of (area × distance from reference axis) divided by total area. y measured from any convenient reference (e.g., top edge).
Watch Out
Do NOT assume the NA is at the geometric center. For a T-beam with a large flange, the NA is in the flange (closer to top). Mistakes in ȳ → wrong I → wrong stresses.
When To Use
FIRST step for any unsymmetric section. Once ȳ is known, distances to top and bottom fibers are c_top and c_bot.
Formula
I = Σ[I_i(centroid) + A_i × d_i²] (parallel-axis theorem)
Meaning
Second moment of inertia about the NA. For each component part: local I about its centroid, plus the parallel-axis term A × d² (d = distance from part centroid to overall NA).
Watch Out
Common error: forgetting the parallel-axis terms and using only Σ I_i(centroid). This gives a much smaller (WRONG) I. Also: d is measured from part centroid to overall NA, NOT from a reference edge.
When To Use
Calculate I for any composite section (T, channel, built-up box, etc.) after locating the NA. This gives the I to use in σ = Mc/I.
Formula
σ_top = (M × c_top) / I; σ_bot = (M × c_bot) / I
Meaning
Top and bottom fiber stresses for unsymmetric section. c_top and c_bot are different, so stresses are different even if M is the same.
Watch Out
For a concrete T-beam with sagging moment: the bottom (tension) stress is larger than the top (compression) stress, so tension steel must be larger. Many students forget which fiber is governed.
When To Use
Design of T-beams and asymmetric sections. The LARGER stress (usually in the tension fiber) governs the design.
Section Title
Unsymmetric Sections (T-Beams, Channels, Angles)
Important Facts
- For unsymmetric sections, c_top ≠ c_bot, so σ_top ≠ σ_bot even under the same bending moment.
- The NA is ALWAYS at the centroid, but for T-beams (heavier flange on top) it is much closer to the top.
- A concrete T-beam in sagging bending has LARGER tension (bottom) than compression (top) stress → reinforce bottom more.
- Locating the NA is the first and most critical step; errors here propagate to I and then to all stresses.
- For a T-beam: if flange is much wider than web, almost all compressive capacity is in the flange, and tensile capacity is in the web.
- The moment of inertia I is always measured about the centroidal (neutral) axis; using an edge or other axis requires parallel-axis correction.
Key Definitions
Term
Neutral Axis (NA) Location
Example
A T-beam with wide flange and thin web: NA is IN the flange, much closer to the top than the centroid of the full envelope.
Definition
Centroidal axis of the section, found by ȳ = Σ(A_i × y_i) / Σ A_i. Not necessarily at mid-depth for unsymmetric sections.
Term
Unsymmetric Section
Example
T-beam, L-angle, channel, Z-section, or composite (steel and timber). Top and bottom fibers experience DIFFERENT stresses under the same moment.
Definition
Cross-section with different shapes or material above and below (or left and right) of a centroidal axis; NA is NOT equidistant from top and bottom.
Diagrams To Know
- T-section with flange and web, showing NA location, c_top, and c_bot.
- Stress distribution for T-beam under sagging moment: compression in flange (smaller), tension in web (larger).
- Centroid calculation diagram: divide section into rectangles, compute centroid of each, then overall centroid.
- Moment of inertia calculation: local I + parallel-axis term for each part.
Formulas
Formula
S_required = M_max / σ_allow
Meaning
Minimum section modulus needed to prevent flexural failure. M_max from the bending-moment diagram, σ_allow from code (NSCP 2015, AISC 360, ACI 318).
Watch Out
This gives a MINIMUM; always pick the next standard size up. Also check that the selected section actually has S ≥ S_required (especially if you round down in mental math).
When To Use
Step 1 of beam design: select a section with S ≥ S_required. Most of the time, the flexure requirement governs.
Formula
τ_max (actual) ≤ τ_allow (code limit)
Meaning
Shear verification: compute actual max shear stress and compare to allowable. If actual ≤ allowable, shear is OK.
Watch Out
Short, heavily loaded beams MAY BE GOVERNED by shear, not flexure. If shear fails, you must widen the section (increase area, not necessarily depth) or use a deeper section (if allowed by deflection limits).
When To Use
Step 2 of beam design: after selecting section for flexure, verify shear does not exceed the allowable. Often shear is not critical in long beams.
Common Values
Value
L/240 (serviceability)
Symbol
Δ_limit
Quantity
Deflection limit (timber floor per NSCP 2015)
Value
L/360 (ordinary floors); L/240 (less critical)
Symbol
Δ_limit
Quantity
Deflection limit (steel beam per AISC 360)
Value
L/360 (general); L/480 (partitions)
Symbol
Δ_limit
Quantity
Deflection limit (concrete per ACI 318)
Section Title
Design Workflow for Beams
Important Facts
- Flexure usually governs (sets section size) for long beams; shear and deflection are checked afterward.
- Short, heavily loaded beams may be governed by shear, deflection, or bearing stress (not bending).
- Always verify BOTH flexure and shear; do not stop after one check.
- Deflection is often more restrictive than stress for long-span beams (e.g., L/240 limit per NSCP 2015).
- Bearing stress at supports must also be checked if the section is small (e.g., narrow timber beam on a steel plate).
- For composite beams (steel and concrete together), the effective section depends on the shear connection and whether composite action is achieved.
Key Definitions
Term
Design Sequence
Example
A 4 m simply supported beam under UDL: calculate reactions, draw SFD/BMD, locate M_max (at midspan), compute S_required, pick a section, verify shear and deflection.
Definition
Typical order: (1) find M_max and V_max from SFD/BMD, (2) size for flexure using S_required = M_max / σ_allow, (3) check shear, (4) check deflection and other serviceability limits.
Diagrams To Know
- Flowchart: Load → SFD/BMD → Find M_max, V_max → S_required → Select section → Check shear → Check deflection → Accept or revise.
- Typical M and V distributions for a simply supported beam under UDL (parabolic M, linear V).
Must Remember
- BENDING STRESS IS LINEAR: σ = My/I, zero at NA, max at extreme fibers. σ_max = M/S where S = I/c. This is the most-tested formula.
- SHEAR STRESS IS PARABOLIC: τ = VQ/(Ib), zero at edges, max at NA. For rectangles: τ_max = 1.5V/A (do NOT forget the 1.5 factor; it's a common trap).
- SECTION MODULUS IS THE DESIGN KEY: S = I/c. Higher S = stiffer in bending. For a rectangle, S = bh²/6, so depth (h) matters far more than width (b) — deep narrow is better than shallow wide.
- FIRST MOMENT Q IS TRICKY: Q = (area above the level) × (distance from NA to centroid of that area). Forgetting this or using the wrong area is the #1 shear-formula mistake on boards.
- UNSYMMETRIC SECTIONS (T-BEAMS): Locate NA first (ȳ = ΣA_i y_i / ΣA_i), then find I using parallel-axis theorem (I_total = Σ[I_local + A × d²]). Top and bottom stresses are DIFFERENT (c_top ≠ c_bot).
- ECCENTRIC LOAD: σ = P/A ± Mc/I where M = P×e. If e > h/6 (outside the kern), TENSION appears. Masonry and footings must stay within the middle third.
- SHEAR FLOW IN BUILT-UP BEAMS: q = VQ/I (force per length through connectors). Spacing s = F/q (nail/bolt capacity divided by flow). The seam at the NA has the HIGHEST flow and needs the CLOSEST spacing.
- SIGN CONVENTION FOR BENDING: Sagging moment (∪ shape) is positive; it puts top fiber in COMPRESSION (−) and bottom in TENSION (+). Hogging (∩) is negative; it reverses the signs.
- DESIGN SEQUENCE: (1) Find M_max and V_max from SFD/BMD. (2) Compute S_required = M_max / σ_allow; pick a section. (3) Verify τ_max = VQ/(Ib) ≤ τ_allow. (4) Check deflection. Most failures come from forgetting step 3 (shear verification).
- COMMON BOARD PITFALLS: Forgetting the 1.5 multiplier for rectangular shear; confusing Q (first moment of detached part) with Q_total; using y (distance to a point) instead of c (to extreme fiber) in σ_max = Mc/I; mixing units in I (mm⁴) and c (mm), getting the wrong answer by a factor of 1000.
Last Minute Tips
- BEFORE EXAM: Memorize σ_max = M/S, τ_max (rect) = 1.5V/A, τ_max (circle) = 4V/(3A), and q = VQ/I. These four formulas appear on nearly every board problem.
- IF STUCK ON SHEAR: Draw the section, shade the area ABOVE your level of interest (that's Q), measure its centroid distance from the NA, multiply A × distance = Q. Then plug into τ = VQ/(Ib). Do NOT guess the 1.5 factor; know it cold for rectangles.
- ON T-BEAMS / UNSYMMETRIC SECTIONS: Always find the NA first (ȳ centroid), then compute I using parallel-axis for each part. THEN compute top and bottom fiber stresses separately (they will differ). The larger stress (usually tension in sagging bending) governs design.
- ECCENTRIC LOAD / SHORT COLUMN: If the problem says 'load applied off-center' or 'eccentric,' immediately think M = P×e and use σ = P/A ± Mc/I. Check if tension appears (σ_min < 0); if it does and the material is masonry or concrete, the design fails.
- ON A TIGHT EXAM: If you have 2 minutes left and see a shear-flow problem on a built-up beam, just remember s = F/q where q = VQ/I. Do NOT try to re-derive it; plug in the numbers, get the spacing, move on. Same for section modulus: S = I/c is the design shortcut.
Comparison Tables
Rows
Values
- LINEAR (0 at NA, max at extreme fibers)
- PARABOLIC for rectangles (0 at top/bottom, max at NA)
Property
Distribution across depth
Values
- Extreme fibers (top and bottom)
- Neutral axis (mid-depth for rectangles)
Property
Maximum location
Values
- At the neutral axis
- At top and bottom surfaces
Property
Zero location
Values
- σ = My/I or σ_max = M/S
- τ = VQ/(Ib) or τ_max = 1.5(V/A) for rectangles
Property
Formula
Values
- Usually GOVERNS; flexure determines section size
- Rarely governs in long beams; important in short, heavy-loaded beams
Property
Design significance
Values
- Excessive curvature, fiber tensile/compressive rupture
- Horizontal slip (rolling shear), web buckling in I-beams
Property
Member failure mode
Columns
- Property
- Bending Stress (σ)
- Shear Stress (τ)
Table Title
Bending Stress vs. Shear Stress Distribution
Rows
Values
- bh²/6
- bh³/12
- c = h/2; most efficient if h >> b
Property
Rectangle (b × h)
Values
- πd³/32
- πd⁴/64
- Same S in all directions (isotropic)
Property
Solid circle (diameter d)
Values
- π(do⁴ − di⁴)/(32do)
- π(do⁴ − di⁴)/64
- Lighter; same stiffness as solid if wall thick enough
Property
Hollow circle (do, di)
Values
- From tables (e.g., 178 × 10⁶ mm³)
- From tables (e.g., 178 × 10⁶ mm⁴)
- Highly efficient; most bending capacity in flanges
Property
Wide-flange (W-section, table value)
Values
- I / c_top or I / c_bot (different top/bottom)
- Sum of components via parallel-axis theorem
- Asymmetric; tension and compression stresses differ
Property
T-section (flange + web)
Columns
- Section Shape
- Formula for S
- Formula for I
- Note
Table Title
Section Modulus (S) for Common Shapes
Rows
Values
- 3V/(2A)
- 1.5 × V/A
- Factor = 1.5; occurs at NA (h/2 from top)
Property
Rectangle (b × h)
Values
- 4V/(3A)
- ≈1.33 × V/A
- Factor = 4/3 ≈ 1.33; at center
Property
Solid circle
Values
- V / A_web (where A_web = t_w × d)
- ≈1.0 × V/A_web
- Most shear in thin web; flanges contribute little
Property
Wide-flange I-beam (approx)
Values
- 1.5 × V/A (same as rectangle)
- 1.5 × V/A
- Always use 1.5 for rectangles, regardless of aspect ratio
Property
Solid rectangular bar (narrow)
Columns
- Section Type
- τ_max Formula
- Equivalent Form (V/A×k)
- Key Point
Table Title
Maximum Shear Stress Factor for Common Sections
Rows
Values
- 4–12 MPa (grade and duration of load)
- 0.8–2.0 MPa (grain orientation)
- NSCP 2015 (Chapter 15: Timber)
Property
Timber (species dependent)
Values
- 0.66 × F_y ≈ 165–180 MPa
- 0.4 × F_y ≈ 100 MPa
- AISC 360-16 (allowable stress design)
Property
Mild steel (Grade 250)
Values
- 0.45 × fc' ≈ 9.5 MPa (compression); flexure governed by steel
- √(fc'/6) ≈ 0.75 MPa (nominal); reinforcement for shear
- ACI 318 (allowable or ultimate limit states)
Property
Reinforced concrete (fc' = 21 MPa typical)
Values
- 50–150 MPa (type and temper)
- 25–100 MPa
- Varies by alloy code
Property
Aluminum alloys
Columns
- Material & Type
- σ_allow (Flexure/Bending)
- τ_allow (Shear)
- Code Reference
Table Title
Allowable Stresses (Typical Codes: NSCP 2015, AISC 360, ACI 318)
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