CELE Structural Theory & Analysis — Analysis of Determinate StructuresCheat Sheet

Formula

DI = (3m + r) - (3n + e_c)

Meaning

m = members, r = reaction components, n = joints (nodes), e_c = equations of condition (internal releases)

Watch Out

Count ALL members (including those that appear as one visual element) and distinguish joints from members. Internal hinge = +1 e_c only; pin connecting k members = (k-1) e_c

When To Use

Classify any planar truss, beam, or frame structure

Formula

DI = r - (3 + e_c)

Meaning

Shortcut for simple beams only: r = total reaction unknowns, 3 = global equilibrium equations, e_c = internal hinges

Watch Out

Only valid for beams, NOT frames. Each internal hinge adds exactly +1 to e_c. Do NOT use for trusses or complex frames.

When To Use

Quick check for determinate beam (DI = 0) or indeterminate (DI > 0) or unstable (DI < 0)

Term

Statically Determinate

Example

Simple supported beam, cantilever beam with one load, three-hinged arch

Definition

DI = 0; reactions and internal forces solvable by equilibrium equations alone (ΣF_x, ΣF_y, ΣM) without using member compatibility.

Term

Statically Indeterminate

Example

Continuous beam, fixed-fixed beam, portal frame with multiple fixed supports

Definition

DI > 0; equilibrium alone is insufficient; compatibility conditions and elastic deformation required to solve.

Term

Unstable Structure

Example

Three collinear supports, reactions parallel to a load direction

Definition

DI < 0 or reactions are concurrent/parallel; cannot resist all possible load directions despite reaction count.

Term

Equation of Condition (e_c)

Example

Internal hinge: ΣM = 0 on one side of hinge; roller release: V = 0 at that point

Definition

Extra equilibrium relation from internal release (hinge, pin, or member release); adds one unknown but also one equation.

Formula

ΣF_x = 0

Meaning

Sum of all horizontal forces (and reaction components) equals zero

Watch Out

Don't forget distributed loads over a length; replace with concentrated equivalent at the centroid. Watch sign: rightward = +, leftward = −

When To Use

Always start here for any structure; rearrange to find horizontal reactions

Formula

ΣF_y = 0

Meaning

Sum of all vertical forces (and reaction components) equals zero

Watch Out

Common error: forgetting that a reaction has BOTH vertical and horizontal components even if only one load direction is given

When To Use

Find vertical reaction components; include weight of members if given

Formula

ΣM = 0 (about any point)

Meaning

Sum of moments about a chosen point equals zero; moment arm = perpendicular distance from force line to point

Watch Out

Choose the moment point wisely to eliminate multiple unknowns. Counterclockwise = + (or reverse, but BE CONSISTENT). Moment of a couple is the same about any point.

When To Use

Take moments about support A to eliminate A's reactions and solve for reactions at other supports

Formula

ΣM_C = 0 (at internal hinge C)

Meaning

At an internal hinge, the net moment on one side = 0 (no moment transfer across hinge)

Watch Out

MUST take moments on ONE side of the hinge only. Apply to the ENTIRE left (or right) portion from the support to the hinge.

When To Use

Three-hinged arch or Gerber beam with internal hinge; use as the 4th equation for determinate structures

Term

Support Reaction

Example

Fixed support: 3 reactions (R_x, R_y, M); pin: 2 reactions (R_x, R_y); roller: 1 reaction perpendicular to surface

Definition

Force (or moment) exerted by a support on the structure to maintain equilibrium; direction opposite to movement prevented.

Term

Simple Support (Pin or Roller)

Example

Beam on knife-edge at each end; bridge simply supported on bearings

Definition

Two reactions (pin) or one reaction perpendicular to surface (roller); no moment transfer.

Term

Fixed Support (Cantilever Base)

Example

Beam rigidly connected to a column base; wall-mounted bracket

Definition

Three reactions: two force components and one moment; prevents all translation and rotation.

Note

Quick check: R_A = R_B = (sum of all vertical loads) / 2

Equation

R_A = ΣF / 2 (for symmetric load on simply supported beam, both ends at same level)

Conditions

Load is symmetric about midspan; no horizontal load; reactions are vertical only

Note

Most common method for finding support reactions in determinate beams

Equation

ΣM_A = 0 ⟹ R_B × L = Σ(P_i × x_i) + w(centroid location)

Conditions

Taking moments about A to solve for B; L = span length

Formula

N (Axial Force) = ΣF_parallel (on one side of section)

Meaning

Sum of forces parallel to member axis; positive = tension, negative = compression (convention may vary — state yours)

Watch Out

Sign convention: establish clearly at the start. Many textbooks use opposite signs for N. Consistency across the diagram is critical.

When To Use

Cut the member at desired section; sum parallel forces on the left (or right, CONSISTENTLY) side

Formula

V (Shear Force) = ΣF_perpendicular (on one side of section)

Meaning

Sum of forces perpendicular to member axis; positive = shear causing clockwise rotation on left face

Watch Out

Shear jumps by the magnitude of the point load at that location. At a distributed load, shear varies linearly.

When To Use

At any section, sum transverse loads on one side. Used to draw shear force diagram (SFD)

Formula

M (Bending Moment) = ΣM_about_section (on one side of section)

Meaning

Sum of moments about the section; positive = tension in bottom fiber (sagging, ∪ shape)

Watch Out

Sign: upward load causes positive (sagging) moment. Negative moment (hogging, ∩) occurs over supports in continuous beams. Slope of M-diagram = shear force (dM/dx = V).

When To Use

Take moments on one side of the cut section to find internal moment. Used to draw moment diagram (BMD)

Term

Axial Force (N)

Example

In a truss member carrying load along its length; in the vertical column of an L-frame

Definition

Internal force parallel to member axis; positive = tension (pulling apart), negative = compression (pushing together).

Term

Shear Force (V)

Example

Maximum at supports of a simply supported beam; zero at midspan of uniformly loaded beam

Definition

Internal force perpendicular to member axis; causes one part to slide past the other; jumps at point loads.

Term

Bending Moment (M)

Example

Maximum at center of simply supported uniformly loaded beam; zero at free ends and over pinned supports

Definition

Internal couple (moment) that resists bending; positive moment creates tension in bottom fiber (sagging parabola).

Term

Sign Convention (Standard)

Example

Uniformly loaded simply supported beam: V positive to the left of center, negative to the right; M is everywhere positive (sagging)

Definition

Positive shear: right face moves up, left face moves down (clockwise on left face). Positive moment: tension in bottom (∪ shape, sagging).

Note

Positive V (upward) on left face means the shear is pulling the left section down and right section up

Equation

V_left = R_A − (sum of loads between A and section) = R_A − ∫w dx

Conditions

For a cantilever or simple beam, taking left-side equilibrium

Note

For cantilever, M increases as you move from free end to fixed end (M_max at fixed support)

Equation

M_section = R_A × x − ∫∫w dx² (for cantilever; add concentrated moment loads where applicable)

Conditions

x measured from left support; integrate distributed load to get moment contribution

Formula

ΣM_crown = 0 (on one side of the crown hinge C)

Meaning

At the crown hinge, net moment on left (or right) side = 0; this gives the horizontal thrust H

Watch Out

Must apply to ONE side of the crown ONLY. Include the weight and all loads on that side, measured perpendicular to the centroidal axis. Forget this condition and you cannot solve the arch.

When To Use

Determine the horizontal thrust in a three-hinged arch (symmetrical or not)

Formula

H = (ΣM_crown_left) / (rise h)

Meaning

H (horizontal thrust) = moment about crown from left side divided by the vertical distance from support to crown

Watch Out

H is the same throughout the arch (any vertical slice). Moment includes the lever arm from the crown to the load.

When To Use

After ΣM_C = 0 and solving for one reaction, use this to find H explicitly

Formula

V_A + V_B = ΣP_vertical (total vertical load)

Meaning

Sum of vertical reactions equals total vertical load (global equilibrium)

Watch Out

Often V_A ≠ V_B even for symmetric arch under point load if load is not at center

When To Use

Check; find V_A or V_B given the other and total load

Formula

H_A + H_B = 0 (for an arch with no external horizontal load at supports)

Meaning

Horizontal reactions are equal and opposite (equilibrium in x-direction)

Watch Out

If there is an external horizontal load P_x, then H_A + H_B = P_x (not zero)

When To Use

Verify that H_A = H_B = H (the thrust)

Term

Three-Hinged Arch

Example

Symmetrical parabolic arch with support hinges at same level and crown hinge at midspan

Definition

Determinate arch with hinged supports at A and B and an internal hinge at the crown C; defines 4 unknowns and has 4 equations (3 global + crown condition).

Term

Horizontal Thrust (H)

Example

In a symmetrical arch under vertical load: H = ΣM_C (at crown from left side) / h, where h = rise

Definition

The horizontal reaction at arch supports; resists the lateral push from the arch curvature; reduces bending compared with a straight beam of same span.

Term

Crown Hinge

Example

In a symmetric arch, often at midspan; allows rotation without moment transfer between left and right halves

Definition

The internal hinge at the top of the arch (or at the apex); enforces the condition ΣM = 0 on each side independently.

Term

Arch Rise (h)

Example

For a parabolic arch spanning 20 m with rise 5 m: h = 5 m

Definition

Vertical distance from support level to the crown hinge; determines the relationship between moment at crown and horizontal thrust.

Note

First step: find vertical reactions using global moment equilibrium

Equation

ΣM_A (global) = 0 ⟹ V_B × L = ΣP_i × x_i

Conditions

L = span length; x_i = horizontal distance of load i from A

Note

This is the crown condition; rearrange to solve for H

Equation

ΣM_C^left = 0 ⟹ V_A × (L/2) − H × h − (loads on left side with their moments) = 0

Conditions

Left side extends from A to C; h = rise (vertical distance from A to C)

Note

Both thrust components are equal; this is the defining characteristic of the three-hinged arch

Equation

ΣF_x = 0 ⟹ H_A = H (from left side condition); H_B = H (by symmetry of moment equilibrium)

Conditions

No external horizontal load at supports

Formula

Analyze suspended spans FIRST; then propagate reactions upward

Meaning

In a Gerber beam, identify the suspended (cantilevered) portion; solve it independently, then add its reaction to the next support below

Watch Out

Many students solve the whole beam at once; you'll get contradictions. Always start with the span that has the fewest supports.

When To Use

Multi-span determinate beams with internal hinges (roller or pin connections between spans)

Formula

e_c (at internal hinge) = number of releases = 1 (for a single hinge between two parts)

Meaning

Each hinge adds one equation of condition (ΣM = 0 at hinge for moment on one side)

Watch Out

Do NOT confuse a simple hinge with a pin connecting k members: a simple internal hinge between two beam segments = 1 e_c; a pin at k members = (k − 1) e_c

When To Use

Count to verify DI = 0 for the compound beam: DI = r − (3 + e_c)

Term

Compound (Gerber) Beam

Example

Main beam rests on three supports; a suspended (secondary) beam connects between two main supports via a hinge

Definition

A determinate multi-span beam with internal hinges (roller or pin) connecting the spans; allows independent analysis of cantilever portions.

Term

Suspended Span

Example

In a three-span beam with a central suspended span: the left and right outer spans are main; the center is suspended on the left support

Definition

The portion of a Gerber beam that is cantilever-like, supported only at one end by another beam; analyzed first, independently.

Term

Internal Hinge (or Pin)

Example

Roller connection between main and secondary beam; hinge pin joining two structural members

Definition

A connection between two adjacent beam segments that allows relative rotation; moment = 0 at the hinge interface.

Note

The reaction of the suspended span becomes a load on the main beam

Equation

Solve suspended span first: ΣM = 0, ΣF = 0 independently

Conditions

Suspended span has loads; find its reactions

Note

This is the most common approach in Philippine board exams

Equation

Add suspended span reaction to main beam; then solve main beam reactions

Conditions

Main beam now has the suspended span reaction as an additional downward load

Term

Right-Hand Rule (for moments)

Example

For a horizontal beam, counterclockwise moment (viewed from above) is positive; causes upward curvature in left overhang

Definition

Thumb in direction of axis (along member); fingers curl from positive to negative; counterclockwise viewed from that direction = positive.

Term

Sagging Moment (Positive)

Example

At midspan of a simply supported uniformly loaded beam

Definition

Bending moment that creates tension in the bottom fiber and compression in the top; forms a ∪ shape (smile).

Term

Hogging Moment (Negative)

Example

Over intermediate supports in a continuous beam

Definition

Bending moment that creates compression in bottom fiber and tension in top; forms a ∩ shape (frown).

Formula

dV/dx = −w (distributed load intensity)

Meaning

Slope of shear diagram = negative of the distributed load; shear decreases if load acts downward

Watch Out

If load is upward (negative w), then dV/dx is positive (shear increases to the right). Watch sign!

When To Use

Verify SFD shape under distributed loads; check for discontinuities at point loads

Formula

dM/dx = V (shear force)

Meaning

Slope of moment diagram = shear force at that section; moment is constant where V = 0

Watch Out

Linear shear → parabolic moment; zero shear → horizontal (constant) moment; positive shear → moment increasing to the right

When To Use

Verify BMD shape; find maximum moment (occurs where V = 0)

Formula

ΔM = ∫V dx (area under SFD)

Meaning

Change in moment between two sections = area under shear force diagram

Watch Out

If SFD is negative, area is negative → moment decreases to the right

When To Use

Quick check: calculate moment at two points and verify difference equals area under SFD

Formula

ΔV = −∫w dx (area under load diagram)

Meaning

Change in shear = negative area under distributed load

Watch Out

Downward load (positive w) → negative change in V as you move right; SFD slopes downward

When To Use

Verify shear diagram shape and values

Term

Inflection Point

Example

Over an interior support of a continuous beam (transition from sagging to hogging)

Definition

Location where bending moment changes sign (from + to − or vice versa); curvature of the beam changes direction.

Term

Critical Section

Example

Midspan of simply supported beam (max M); just left of a support (max V); at a point load (shear jump)

Definition

A section where internal forces reach local maximum or minimum; typically where V = 0 for moment, or at supports and point loads.

Formula

DI = (3m + r) − (3n + e_c)

Meaning

For frames: m = number of members, r = reaction components, n = joints, e_c = internal releases

Watch Out

Count EVERY member (horizontal, vertical, diagonal); a single L-corner counts as TWO members meeting at a joint. A pin at a joint where 3 members meet = 2 e_c, not 1.

When To Use

Determine if a multi-member frame is determinate, indeterminate, or unstable

Term

Portal Frame (Rectangular Frame)

Example

Building frame: four members (two columns, one beam at top, base connections); base is fixed or pinned

Definition

A rigid frame with vertical columns and a horizontal beam, typically cantilevered or supported on a spread foundation.

Term

Knee (or Corner Joint)

Example

Where a column meets a beam in a portal; moment at knee on column = moment at knee on beam (internal equilibrium)

Definition

A rigid joint where two members meet at an angle (typically 90° in portal frames); moment transfer occurs.

Term

Member Axis (Local Coordinates)

Example

In a vertical column: axial = vertical, shear = horizontal; in a horizontal beam: axial = horizontal, shear = vertical

Definition

For each member of a frame, a local coordinate system: axial force along the member, shear perpendicular, moment about the member.

Note

Careful with sign: forces on the member face pointing outward from joint are INTERNAL forces, equal and opposite on the joint free body

Equation

Joint equilibrium: ΣF_x = 0, ΣF_y = 0, ΣM = 0 (at each joint of a frame)

Conditions

Must apply to every joint; internal forces on cut members balance with applied loads and reactions

Note

Often simpler than joint-by-joint analysis; typical in board exams

Equation

Section method: cut the frame and apply equilibrium to one part

Conditions

Choose a section that cuts through the member where you want internal forces

Values

• DI = 0
• Yes, use ΣF = 0, ΣM = 0 only
• Yes; concurrent/parallel reactions = unstable
• Simple beam, cantilever, three-hinged arch

Property

Statically Determinate

Values

• DI > 0
• No; need compatibility + elastic constants
• Yes; indeterminate degree must match equation count
• Continuous beam, fixed-fixed beam, two-story frame

Property

Statically Indeterminate

Values

• DI < 0 OR reactions concurrent/parallel
• Cannot solve; structure can move freely
• Structure will fail or displace without support
• Three collinear supports, or all reactions parallel to a load

Property

Unstable (Mechanism)

Values

• DI = 0 (by design with internal hinges)
• Yes, but use internal hinge condition first
• Yes; hinges allow cantilever action safely
• Gerber beam: suspended span + main spans with hinges

Property

Compound Beam

Values

• 2
• R_x (horizontal), R_y (vertical)
• No (M = 0)
• Circle or •

Property

Pin (Hinge)

Values

• 1
• R perpendicular to surface (usually vertical)
• No (M = 0)
• Circle on wheels or ▲ with wheels

Property

Roller

Values

• 3
• R_x, R_y, and M (moment)
• Yes; can be large
• Thick line or filled square

Property

Fixed (Cantilever)

Values

• Counted as e_c = 1 (adds one equation)
• Forces transmitted; moment BLOCKED
• No; ΣM = 0 on each side
• Small circle, dashed line, or pin symbol between members

Property

Internal Hinge

Values

• Tension (pulling)
• Pointing right (+)
• Pointing left, magnitude N
• Pointing right, magnitude N
• Often zero in beams; non-zero in trusses & frames

Property

Axial Force (N)

Values

• Upward on left face, downward on right (clockwise)
• Upward, magnitude V (internal shear resisting slip)
• Downward, magnitude V
• Positive from left support toward midspan; negative from midspan toward right support

Property

Shear Force (V)

Values

• Sagging (∪ shape, bottom in tension)
• Counterclockwise at cut, magnitude M
• Clockwise at cut, magnitude M
• Zero at supports; maximum positive at midspan under uniform load

Property

Bending Moment (M)

Values

• dV/dx = 0 (constant)
• dM/dx = V (linear if V ≠ 0)
• Horizontal (constant)
• Linear (sloped line)

Property

No distributed load (w = 0)

Values

• dV/dx = −w (decreases left to right)
• dM/dx = V (linear shear → parabolic moment)
• Sloped downward (linear decrease)
• Parabola opening down (concave)

Property

Uniform downward load (w > 0)

Values

• Jump down by P (discontinuity)
• Slope changes abruptly (corner in BMD)
• Vertical drop equal to P
• Kink (change in slope) at load location

Property

Concentrated point load P downward

Values

• No change (ΔV = 0)
• Jump up by M (discontinuity)
• No change (constant between loads)
• Vertical jump equal to M

Property

Concentrated moment M_applied

Values

• DI = 0 (determinate)
• DI = 0 (determinate; 4 unknowns, 4 equations including crown condition)

Property

Determinacy

Values

• V_A = ΣP · (L − x) / L; V_B = ΣP · x / L (beam equation)
• V_A and V_B found from global ΣM = 0, same as beam

Property

Vertical Reactions

Values

• None (H = 0) if no horizontal load
• H_A = H_B = H (non-zero thrust due to curvature); found from crown condition

Property

Horizontal Reactions

Values

• M_max = P · L / 4 (for point load at center)
• M_max ≤ M_beam − H × h (reduced by the thrust effect)

Property

Maximum Bending Moment

Values

• Simple design, easy to analyze
• Lower internal bending moment; more efficient for long spans (less material, less deflection)

Property

Primary Advantage

Values

• No crown; simply ΣM = 0 everywhere for equilibrium
• ΣM_C = 0 (on one side of crown hinge); essential for solving H

Property

Crown Condition