Analysis and Design of Beams (Flexure)

Analysis and Design of Beams (Flexure)

$Learning Objectives$

Apply equilibrium and strain compatibility to rectangular reinforced-concrete sections at nominal flexural strength.
Use the Whitney equivalent rectangular compression block with the correct $\beta_1$ limits.
Verify rather than blindly assume whether tension and compression reinforcement has yielded.
Calculate $M_n$ , $\phi$ , and $\phi M_n$ for singly and doubly reinforced beams.
Distinguish minimum reinforcement, the beam minimum tensile-strain requirement, balanced behavior, and the tension-controlled limit.
Analyze flanged beams using an effective compression-flange width and the correct compression-block case.
Distinguish ordinary beam flexure, side-face skin reinforcement, and deep-beam/strut-and-tie behavior.
Check reinforcement geometry, effective depth, extreme-tension depth, and constructability assumptions before accepting a design.

$Flexural nominal strength$

The moment resistance $M_n$ of a section evaluated at the nominal-strength strain state using equilibrium, compatibility, the specified material models, and the applicable reinforced-concrete strength provisions.

$NSCP 2015 / Adopted ACI 318-14 Basis$

This topic uses the Philippine NSCP 2015 concrete-design context and its adopted ACI 318-14 flexural basis unless a different edition is explicitly identified. The course therefore uses an extreme concrete compression strain of $0.003$ , the ACI 318-14 rectangular stress block, and the ACI 318-14 strain-based $\phi$ limits described below. Newer ACI editions must not be mixed silently into these calculations.

$Core Flexural Analysis Assumptions$

STEP-BY-STEP

Sections that are plane before bending remain plane after bending, so longitudinal strain varies linearly over the section depth.
Reinforcement strain equals the strain of the surrounding concrete at the same level when adequate bond and anchorage are present.
Internal compression and tension forces satisfy equilibrium.
Concrete tensile stress is neglected when calculating nominal flexural strength after cracking.
The extreme concrete compression strain at nominal flexural strength is taken as $\epsilon_{cu}=0.003$ .
Concrete compression is represented by the equivalent rectangular stress block defined by the adopted code basis.
Reinforcement stress must be obtained from the calculated reinforcement strain; $f_s=f_y$ is valid only after yielding has been verified.

$d$

The distance from the extreme compression fiber to the centroid of the longitudinal tension reinforcement.

$d_t$

The distance from the extreme compression fiber to the center of the reinforcement layer farthest into the tension zone. It is the depth used to evaluate the net tensile strain $\epsilon_t$ for beam minimum-strain checks, strain classification, and the flexural strength-reduction factor.

$Do Not Confuse d with d_t in Multilayer Reinforcement$

For one tension-reinforcement layer, $d=d_t$ , so the distinction is invisible. For multiple tension layers, $d$ is the depth to the area centroid of the longitudinal tension reinforcement, while $d_t$ is the depth to the extreme tension layer. They generally differ. Calculate each layer strain from its own depth. If the layer stresses differ, the internal tension-force resultant does not generally act at $d$ ; calculate the layer forces and their moments individually. If all participating layers develop the same stress, such as the same yielded $f_y$ , the force resultant coincides with the steel-area centroid at $d$ . Never use the area centroid as a substitute for the extreme-layer strain.

$Whitney equivalent rectangular stress block$

The code-approved equivalent concrete compression block having uniform stress $0.85f'_c$ over depth $a=\beta_1c$ , where $c$ is the neutral-axis depth measured from the extreme compression fiber.

Equivalent Compression-Block Depth

Relates the equivalent rectangular block depth to the neutral-axis depth.

a=\beta_1c

Variables

Symbol	Description	Unit
$a$	Equivalent rectangular stress-block depth	mm
$\beta_1$	Stress-block depth factor	-
$c$	Neutral-axis depth from the extreme compression fiber	mm

$\beta_1$

For normal-strength ranges used in this course, $\beta_1=0.85$ for $17\le f'_c\le28\,\text{MPa}$ . For $f'_c>28\,\text{MPa}$ , reduce $\beta_1$ by $0.05$ for each $7\,\text{MPa}$ increase above $28\,\text{MPa}$ , but do not take $\beta_1<0.65$ .

\beta_1=\max\left[0.65,\;0.85-0.05\left(\frac{f'_c-28}{7}\right)\right]

The reduction applies only above $28\,\text{MPa}$ ; do not extrapolate the line below the code range.

$Singly Reinforced Rectangular Beams$

For a rectangular section with tension steel only, the concrete resultant is

$C_c=0.85f'_cba$ ,

acting at $a/2$ from the compression face. The tension force is $T=\sum A_{si}f_{si}$ . For a single yielded tension layer this reduces to $T=A_sf_y$ . Equilibrium requires $C_c=T$ . If a yielded-steel shortcut is proposed for multiple tension layers, verify the strain and stress of every participating layer before combining their forces. If any layer remains elastic, use its actual $f_{si}=E_s\epsilon_{si}$ rather than forcing the entire tension reinforcement to $f_y$ .

Singly Reinforced Equilibrium When Tension Steel Yields

Closed-form compression-block depth after the assumption $f_s=f_y$ has been verified for the participating tension reinforcement.

a=\frac{A_sf_y}{0.85f'_cb}

Variables

Symbol	Description	Unit
$A_s$	Area of tension reinforcement verified to be at the same yield stress	$mm^2$
$f_y$	Specified reinforcement yield strength	MPa
$f'_c$	Specified concrete compressive strength	MPa
$b$	Compression-face width of the rectangular section	mm

Extreme Tension-Steel Strain from Compatibility

Net tensile strain at nominal strength in the reinforcement layer farthest from the extreme compression fiber.

\epsilon_t=0.003\left(\frac{d_t-c}{c}\right)

Variables

Symbol	Description	Unit
$\epsilon_t$	Net tensile strain in the extreme tension reinforcement at nominal strength	-
$d_t$	Depth from the extreme compression fiber to the extreme tension-reinforcement layer	mm
$c$	Neutral-axis depth	mm

$Layer-by-Layer Compatibility$

For any tension layer $i$ at depth $d_i$ , calculate

\epsilon_{si}=0.003\left(\frac{d_i-c}{c}\right)

and obtain $f_{si}$ from the reinforcement stress-strain model. The extreme value used for $\epsilon_t$ corresponds to $d_t=\max(d_i)$ on the tension side. If all participating tension layers have yielded to the same $f_y$ , their forces may be combined at the steel-area centroid $d$ for the moment calculation. If their stresses differ, retain the layer forces separately rather than forcing their resultant to act at $d$ . Neither case changes the requirement to use $d_t$ for strain classification.

Nominal Moment of a Yielded Singly Reinforced Rectangular Section

For one tension layer, or multiple tension bars at the same stress, the tension resultant acts at the steel-area centroid d.

M_n=A_sf_y\left(d-\frac{a}{2}\right)

Variables

Symbol	Description	Unit
$M_n$	Nominal flexural strength	N mm
$A_s$	Area of yielded tension reinforcement at the common stress fy	$mm^2$
$f_y$	Specified reinforcement yield strength	MPa
$d$	Depth to the area centroid of the longitudinal tension reinforcement	mm
$a$	Equivalent stress-block depth	mm

$Do Not Use the Yielded-Steel Formula Before Checking Yield$

The equation $a=A_sf_y/(0.85f'_cb)$ assumes the participating tension steel represented by $A_s$ has reached the same $f_y$ . For one tension layer, calculate $c=a/\beta_1$ and its strain immediately. For multiple layers, calculate every $\epsilon_{si}$ at its own $d_i$ and verify every layer included in $A_sf_y$ has yielded. If any layer has $\epsilon_{si}<\epsilon_y$ , discard the lumped yielded-steel assumption and solve equilibrium with the actual layer stresses. Separately, use the extreme layer at $d_t$ to determine $\epsilon_t$ for the beam minimum-strain check and $\phi$ classification.

$Balanced strain condition$

The theoretical condition at which the extreme compression concrete reaches $0.003$ at the same time the tension reinforcement reaches its yield strain $\epsilon_y$ .

Balanced Reinforcement Ratio for the Idealized Yield Model

Useful reference ratio obtained from balanced strain compatibility for ordinary nonprestressed reinforcement.

\rho_b=0.85\beta_1\frac{f'_c}{f_y}\left(\frac{600}{600+f_y}\right)

Variables

Symbol	Description	Unit
$\rho_b$	Balanced tension-reinforcement ratio	-
$\beta_1$	Stress-block depth factor	-
$f'_c$	Specified concrete compressive strength	MPa
$f_y$	Specified reinforcement yield strength	MPa

$Minimum Reinforcement and Ductility Are Different Checks$

Minimum tension reinforcement is intended to prevent an abrupt loss of flexural resistance immediately after concrete cracking. It is not the same as a maximum reinforcement ratio, balanced ratio, or tension-controlled classification. For ordinary nonprestressed rectangular beams on this course basis, evaluate the code minimum area and separately verify the strain limits at nominal strength.

Minimum Tension Reinforcement for Ordinary Nonprestressed Beams

Use the greater of the two ACI/NSCP expressions when the stated beam provision applies.

A_{s,\min}=\max\left(\frac{0.25\sqrt{f'_c}}{f_y}b_wd,\;\frac{1.4}{f_y}b_wd\right)

Variables

Symbol	Description	Unit
$A_{s,\min}$	Minimum required tension-reinforcement area	$mm^2$
$b_w$	Web width	mm
$d$	Effective depth to the centroid of the longitudinal tension reinforcement	mm

$Beam Minimum Tensile Strain versus Tension-Controlled Classification$

For the adopted ACI 318-14 basis, nonprestressed flexural members with negligible axial compression are generally required to have $\epsilon_t\ge0.004$ at nominal strength. This is a minimum beam ductility requirement, not the definition of a tension-controlled section.

A section is tension-controlled when $\epsilon_t\ge0.005$ , which permits $\phi=0.90$ for flexure. Between the yield-strain boundary and $0.005$ , $\phi$ lies in the transition region. In multilayer reinforcement, these limits apply to the extreme tension reinforcement at $d_t$ , not automatically to the centroidal depth $d$ . Do not use $0.004$ and $0.005$ interchangeably.

Strength-Reduction Factor for Nonprestressed Flexure on the Adopted Basis

Strain-based phi rule for tied/nonprestressed flexural sections using the actual yield strain.

\phi= \begin{cases} 0.65, & \epsilon_t\le\epsilon_y\\ 0.65+0.25\dfrac{\epsilon_t-\epsilon_y}{0.005-\epsilon_y}, & \epsilon_y<\epsilon_t<0.005\\ 0.90, & \epsilon_t\ge0.005 \end{cases}

Variables

Symbol	Description	Unit
$\phi$	Strength-reduction factor	-
$\epsilon_t$	Extreme tension-steel strain at nominal strength	-
$\epsilon_y$	Reinforcement yield strain, fy/Es	-

$Interactive Section Verification$

Use the section analyzer to change $f'_c$ , $f_y$ , section dimensions, and reinforcement. Watch the computed yield strain, neutral axis, steel stress, strain classification, $M_n$ , and $\phi M_n$ . The interactive model uses one tension layer, so its displayed $d$ is also $d_t$ . Invalid bar layouts are flagged rather than treated as legitimate designs.

RC Beam Section Analysis

NSCP 2015 / adopted ACI 318-14 rectangular-section model. Reinforcement stress is solved from the actual strain state.

Beam width

b

300 mm

Beam height

h

500 mm

Tension-steel centroid from tension face60 mm

f'_c

28 MPa

f_y

420 MPa

Longitudinal bar diameter20 mm

Number of tension bars3

Section results

d

440.0 mm

A_s

942 mm²

\beta_1

0.850

c

65.2 mm

a

55.4 mm

\epsilon_y

0.00210

\epsilon_t

0.01724

f_s

420.0 MPa

Steel state

Yielded

Strain class

Tension-controlled

\phi

0.900

Clear bar spacing

80.0 mm

M_n

163.20 kN·m

\phi M_n

146.88 kN·m

Scope: singly reinforced rectangular flexure only. The horizontal fit check uses 40 mm side clear cover and a simplified clear-spacing criterion of max(25 mm, bar diameter); aggregate-size, bundling, development, shear, serviceability, and seismic detailing are outside this visualizer.

$Doubly reinforced beam$

A beam containing both tension and compression longitudinal reinforcement, commonly used when section dimensions are constrained, moment demand is high, moment reversal is expected, or compression steel is otherwise required by structural detailing.

$Doubly Reinforced Force Equilibrium$

Compression steel does not automatically yield. Its strain is obtained from the same linear strain diagram:

$\epsilon'_s=0.003(c-d')/c$ ,

and its stress is limited by the reinforcement constitutive model. When compression reinforcement lies inside the equivalent rectangular concrete block, one consistent bookkeeping convention is

$C_c=0.85f'_cb\beta_1c$

plus the net compression-steel contribution

$C'_s=A'_s(f'_s-0.85f'_c)$ .

The subtraction prevents double-counting the concrete volume displaced by the steel. An alternative convention may exclude the displaced concrete from $C_c$ explicitly; either convention is acceptable only if it is applied consistently.

Doubly Reinforced Equilibrium with Compression Steel inside the Stress Block

Consistent equilibrium equation using the net compression-steel contribution for a single equivalent tension-steel stress.

0.85f'_cb\beta_1c+A'_s(f'_s-0.85f'_c)=A_sf_s

Variables

Symbol	Description	Unit
$A'_s$	Compression-reinforcement area	$mm^2$
$f'_s$	Compression-reinforcement stress from compatibility	MPa
$A_s$	Tension-reinforcement area represented by the common stress fs	$mm^2$
$f_s$	Common tension-reinforcement stress after compatibility verification	MPa

Doubly Reinforced Nominal Moment with the Net-Steel Convention

Applicable when the tension reinforcement can be represented by one resultant at its area centroid d, such as one layer or equal-stress yielded layers.

M_n=C_c\left(d-\frac{a}{2}\right)+C'_s(d-d')

Variables

Symbol	Description	Unit
$C_c$	Concrete compression resultant	N
$C'_s$	Net compression-steel resultant	N
$d$	Depth to the area centroid of the represented tension reinforcement	mm
$d'$	Depth from the compression face to compression-steel centroid	mm

$General Multilayer Tension Reinforcement$

If tension layers develop different stresses, do not force them into the preceding $A_sf_s$ and $d$ representation. Satisfy equilibrium with $T=\sum A_{si}f_{si}$ and calculate nominal moment from the actual concrete, compression-steel, and individual tension-layer forces and lever arms. The code-defined effective depth $d$ remains the steel-area centroid, while the force-resultant location is a separate mechanics quantity.

$Compression Steel Yield Must Be Verified$

Assuming $f'_s=f_y$ can materially distort the neutral-axis solution and moment capacity. Calculate $\epsilon'_s$ from the final $c$ , evaluate $f'_s=E_s\epsilon'_s$ while the steel is elastic, cap it at $f_y$ when yielding occurs, and then re-establish equilibrium. A solved value of $c$ is not valid if the stress assumptions used to obtain it contradict the resulting strains.

$Flanged beam$

A beam whose monolithic slab participates as part of the compression flange over an effective width permitted by the code, producing T- or L-shaped compression geometry when the flange is in compression.

$T-Beam and L-Beam Analysis$

First determine the effective flange width from the applicable NSCP/ACI limits. Then assume the compression block lies within the flange and calculate $a$ . If $a\le h_f$ , analyze the section as a rectangular section of width $b_f$ . If $a>h_f$ , split the concrete compression into the flange overhang and the web so that equilibrium and the centroid of compression are calculated from the actual equivalent block geometry.

Effective flange-width rules depend on beam location, span, flange thickness, and spacing to adjacent webs. They are code limits—not permission to use the entire slab width by default.

$Deep beam$

A member or region in which the load and support geometry produces significant nonlinear strain distribution and direct compression-strut action, so ordinary beam flexure assumptions are not adequate.

$Deep-Beam Behavior Is Not a Skin-Reinforcement Rule$

On the adopted basis, a deep beam is identified from span-to-depth and load-to-support geometry, including regions where concentrated loads occur within about twice the member depth from a support and direct strut action develops. Such regions are designed using the applicable deep-beam provisions and commonly the strut-and-tie method.

Side-face skin reinforcement is a separate ordinary-beam detailing requirement. For sufficiently deep flexural members, longitudinal reinforcement is distributed along the side faces near the tension zone to control crack widths. A beam does not become a deep beam merely because its overall depth exceeds the threshold that triggers skin reinforcement, and deep-beam web reinforcement should not be described as ordinary skin reinforcement.

$Do Not Apply Plane-Sections Flexure to a Deep-Beam D-Region$

Where the strain field is strongly disturbed by nearby concentrated loads, supports, openings, or geometric discontinuities, the ordinary linear strain distribution used for slender beam flexure may be invalid. Use the governing deep-beam or strut-and-tie provisions rather than forcing a familiar $M_n=A_sf_y(d-a/2)$ calculation onto a discontinuity region.

$One-Way Joist and Ribbed Systems$

A one-way joist system consists of regularly spaced ribs and a monolithic top slab. Qualification for special joist provisions depends on geometric limits such as minimum rib width, maximum depth-to-width ratio, and maximum clear spacing. If those limits are not satisfied, the ribs must be designed under the ordinary provisions applicable to their actual geometry rather than assuming special joist allowances.

$Reinforcement Geometry Is Part of the Design$

Calculated steel area is not a complete design. Selected bars must fit inside the stirrups and concrete cover with code-compliant clear spacing, practical layer arrangement, and a steel-area centroid consistent with the $d$ used by code expressions. Compression bars must be laterally supported as required. When bars are arranged in multiple tension layers, recalculate $d$ , identify $d_t$ separately, and calculate each layer strain. If their stresses differ, calculate individual layer forces rather than treating $d$ as the force-resultant location.

$Interactive Reinforcement-Geometry Check$

Use the 3D viewer to change beam dimensions, clear cover, stirrup diameter and spacing, longitudinal bar diameter, and bar quantity. The viewer checks simplified clear-spacing and cover geometry and suppresses reinforcement cages that do not physically fit. It is a detailing visualizer, not a substitute for the complete code spacing, aggregate-size, development-length, splice, or seismic-detailing checks.

RC Beam Reinforcement Geometry

Adjust the section and cage geometry. Invalid cover or bar-spacing states are flagged instead of rendered as acceptable reinforcement.

Geometry valid

Beam width300 mm

Beam height500 mm

Beam length3000 mm

Clear cover to stirrup40 mm

Top bars2

Bottom bars3

Longitudinal bar diameter20 mm

Stirrup diameter10 mm

Maximum stirrup spacing200 mm

300 × 500 mm section · 3000 mm long

Cover 40 mm · stirrups Ø10 @ 194 mm actual

Top clear spacing 160 mm · bottom 70 mm

Drag to rotate · scroll/pinch to zoom

Spacing check shown here is a simplified geometric screen using clear spacing ≥ max(bar diameter, 25 mm). Project detailing must also satisfy the governing code provisions, aggregate-size effects, development, splice, and constructability requirements.

The viewer is a reinforcement-geometry teaching model, not a full structural designer. Bar anchorage, hooks, seismic detailing, and member strength are outside this component’s scope.

3D Flexural-Response Studio: Strain Compatibility and Internal Forces

$Nominal-Strength 3D Model — NSCP 2015 / Adopted ACI 318-14$

The 3D flexural-response studio uses the same one-layer rectangular-beam strain-compatibility basis as the section analyzer above. It solves the neutral axis, Whitney block, actual tension-steel strain/stress, $M_n$ , strain-based $\phi$ , and $\phi M_n$ without assuming steel yield before checking $\epsilon_t\ge\epsilon_y$ . The concrete compression block, neutral-axis plane, tension bars, and $C_c$ – $T$ force couple are generated from that same state. The displayed member curvature and crack pattern are schematic demand cues only; they are not service-deflection or crack-width predictions.

Modeled singly reinforced flexural state satisfies the displayed lesson checks.

Geometry, minimum steel, adopted minimum tensile strain, and Mu ≤ φMn are satisfied for this one-layer rectangular-beam teaching model.

Member curvature and crack graphics are schematic, scaled by Mu/φMn and the 8× visual factor. They are not service-deflection or crack-width predictions. Capacity, strain, stress, force, and φ results remain quantitative.

Use the guided sequence to trace equilibrium, strain compatibility, steel yield verification, and the final φMn capacity check.

300 mm

500 mm

60 mm

28 MPa

420 MPa

4.5 m

80%

8×

Ø20 mm

Strain compatibility

β1: 0.850
c: 65.22 mm
a: 55.44 mm
εy: 0.00210
εt: 0.01724
fs: 420.0 MPa
Steel state: Yielded
Class: Tension-controlled

Strength / demand

As: 942 mm²
As,min: 440 mm²
Mn: 163.20 kN·m
φ: 0.900
φMn: 146.88 kN·m
Mu: 117.50 kN·m
Equivalent center P: 104.4 kN
C−T closure: 0.00e+0 N

$Professional Flexural Analysis Checklist$

STEP-BY-STEP

Establish the code edition and material strengths before selecting coefficients.
Determine $b$ , $h$ , clear cover, transverse reinforcement, the actual tension-layer depths $d_i$ , the steel-area centroid $d$ , the extreme tension-reinforcement depth $d_t$ , and any $d'$ from actual geometry.
Calculate $\beta_1$ from $f'_c$ using the adopted edition.
Write equilibrium and compatibility before assuming reinforcement has yielded.
Solve for $c$ , then compute $a=\beta_1c$ and every reinforcement-layer strain and stress from its actual depth.
Revisit the solution if any assumed yielded/elastic steel state is contradicted by the calculated strain.
Compute $M_n$ from the actual internal resultants and lever arms. Combine tension layers at $d$ only when their equal stresses make the force resultant coincide with the area centroid; otherwise retain the layer forces separately.
Determine $\epsilon_t$ from the extreme tension layer at $d_t$ , determine $\epsilon_y$ , then perform the beam minimum-strain check, strain classification, and $\phi$ calculation.
Verify $\phi M_n\ge M_u$ and minimum reinforcement.
Confirm selected bars, spacing, cover, anchorage, development, and serviceability separately.

Key Takeaways

Flexural strength is governed by equilibrium plus strain compatibility; assuming every reinforcing bar is at $f_y$ is not a valid general method.
The adopted stress block uses $a=\beta_1c$ , with $\beta_1$ reducing above $28\,\text{MPa}$ to a minimum of $0.65$ .
For a single yielded tension layer, the familiar $a=A_sf_y/(0.85f'_cb)$ and $M_n=A_sf_y(d-a/2)$ apply directly; multiple equal-stress yielded layers may also be combined at their area centroid $d$ .
In multilayer reinforcement, $d$ is the steel-area centroid defined by the section geometry, while $d_t$ locates the extreme tension reinforcement used for $\epsilon_t$ , minimum-strain checks, strain classification, and $\phi$ . If layer stresses differ, calculate the force resultant separately from $d$ .
Minimum reinforcement, balanced behavior, $\epsilon_t\ge0.004$ , and tension-controlled $\epsilon_t\ge0.005$ are distinct concepts on the NSCP 2015 / ACI 318-14 basis.
The transition $\phi$ calculation uses the actual reinforcement yield strain $\epsilon_y=f_y/E_s$ ; it must respond when $f_y$ changes.
Doubly reinforced analysis must solve compression-steel strain and stress rather than assuming compression-steel yield.
Deep-beam/strut-and-tie behavior is distinct from ordinary side-face skin reinforcement for deep flexural members.
A reinforcement selection is acceptable only when its actual geometry, layer strains, steel-area centroid, extreme-tension depth, internal force resultants, and $\phi M_n$ satisfy the design—not merely when a theoretical required steel area has been computed.

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Learning Objectives

Flexural nominal strength

NSCP 2015 / Adopted ACI 318-14 Basis

Core Flexural Analysis Assumptions

Effective depth (ddd)

Extreme tension-reinforcement depth (dtd_tdt​)

Do Not Confuse d with d_t in Multilayer Reinforcement

Whitney equivalent rectangular stress block

Equivalent Compression-Block Depth

β1\beta_1β1​ for the Adopted ACI 318-14 Basis

Singly Reinforced Rectangular Beams

Singly Reinforced Equilibrium When Tension Steel Yields

Extreme Tension-Steel Strain from Compatibility

Layer-by-Layer Compatibility

Nominal Moment of a Yielded Singly Reinforced Rectangular Section

Do Not Use the Yielded-Steel Formula Before Checking Yield

Balanced strain condition

Balanced Reinforcement Ratio for the Idealized Yield Model

Minimum Reinforcement and Ductility Are Different Checks

Minimum Tension Reinforcement for Ordinary Nonprestressed Beams

Beam Minimum Tensile Strain versus Tension-Controlled Classification

Strength-Reduction Factor for Nonprestressed Flexure on the Adopted Basis

Interactive Section Verification

RC Beam Section Analysis

Section results

Doubly reinforced beam

Doubly Reinforced Force Equilibrium

Doubly Reinforced Equilibrium with Compression Steel inside the Stress Block

Doubly Reinforced Nominal Moment with the Net-Steel Convention

General Multilayer Tension Reinforcement

Compression Steel Yield Must Be Verified

Flanged beam

T-Beam and L-Beam Analysis

Deep beam

Deep-Beam Behavior Is Not a Skin-Reinforcement Rule

Do Not Apply Plane-Sections Flexure to a Deep-Beam D-Region

One-Way Joist and Ribbed Systems

Reinforcement Geometry Is Part of the Design

Interactive Reinforcement-Geometry Check

RC Beam Reinforcement Geometry

3D Flexural-Response Studio: Strain Compatibility and Internal Forces

Nominal-Strength 3D Model — NSCP 2015 / Adopted ACI 318-14

Professional Flexural Analysis Checklist

$Learning Objectives$

$Flexural nominal strength$

$NSCP 2015 / Adopted ACI 318-14 Basis$

$Core Flexural Analysis Assumptions$

$d$

$d_t$

$Do Not Confuse d with d_t in Multilayer Reinforcement$

$Whitney equivalent rectangular stress block$

$\beta_1$

$Singly Reinforced Rectangular Beams$

$Layer-by-Layer Compatibility$

$Do Not Use the Yielded-Steel Formula Before Checking Yield$

$Balanced strain condition$

$Minimum Reinforcement and Ductility Are Different Checks$

$Beam Minimum Tensile Strain versus Tension-Controlled Classification$

$Interactive Section Verification$

$Doubly reinforced beam$

$Doubly Reinforced Force Equilibrium$

$General Multilayer Tension Reinforcement$

$Compression Steel Yield Must Be Verified$

$Flanged beam$

$T-Beam and L-Beam Analysis$

$Deep beam$

$Deep-Beam Behavior Is Not a Skin-Reinforcement Rule$

$Do Not Apply Plane-Sections Flexure to a Deep-Beam D-Region$

$One-Way Joist and Ribbed Systems$

$Reinforcement Geometry Is Part of the Design$

$Interactive Reinforcement-Geometry Check$

$Nominal-Strength 3D Model — NSCP 2015 / Adopted ACI 318-14$

$Professional Flexural Analysis Checklist$