Structural engineering

This chapter covers the following:

• Stability and determinacy of structures
• Snow and wind loads
• Load combinations for steel member design: ASD vs LRFD
• Tension member design: ASD vs LRFD
• Compression member design: ASD vs LRFD
• Flexural member design: ASD vs LRFD
• Shear member design: ASD vs LRFD

Stability and determinacy of structures

Stability and determinacy tell you whether a structure can be analyzed using statics and whether it will hold its shape under load.

Stability

Definitions

A structure is stable if it maintains its shape and position under applied loads and does not undergo rigid body motion (translation or rotation).

A stable structure resists rigid body motion (translation or rotation) and also avoids internal collapse.

• Externally stable: The entire structure doesn’t move as a rigid body.
• Internally stable: The members are arranged so the structure can’t collapse internally.

Determinacy

Definitions

A structure is statically determinate if all support reactions and internal forces can be found using only the equations of static equilibrium.

A statically determinate structure has just enough unknowns that the equilibrium equations are sufficient to solve for all reactions and internal forces.

Equilibrium equations (2D)

$$\sum F_x=0,\sum F_y=0,\sum M=0$$

Equilibrium equations (3D)

$$\sum F_x=0,\sum F_y=0,\sum F_z=0,\sum M_x=0,\sum M_y=0,\sum M_z=0$$

Determinacy and stability criteria

Beams (2D)

For a simply supported beam:

• Number of unknown support reactions: 3 (1 pin + 1 roller)
• Number of equilibrium equations: 3

$$\text{If }r=3\Rightarrow \text{Statically Determinate}$$

If $r>3$ $\to$ Indeterminate If $r<3$ $\to$ Unstable

Frames (2D)

Let:

• $r$: number of unknown reactions
• $m$: number of members
• $j$: number of joints

Determinacy condition:

$$r+m=3j$$

• If $r+m=3j$: Statically determinate
• If $r+m>3j$: Indeterminate
• If $r+m<3j$: Unstable

Trusses (2D)

Let:

• $m$: number of members
• $j$: number of joints
• $r$: number of support reactions

Determinacy condition:

$$m+r=2j$$

• If $m+r=2j$: Statically determinate and stable
• If $m+r<2j$: Unstable
• If $m+r>2j$: Indeterminate

Examples

Example 1: Truss

Given:

• $m=3$ members
• $j=3$ joints
• $r=3$ reactions

Check determinacy:

$$\begin{gathered} m+r=3+3=6 \\ 2j=2\times 3=6 \end{gathered}$$

Statistically determinate and stable

Example 2: Frame

Given:

• $m=5$ members
• $j=4$ joints
• $r=3$ reactions

Check:

$$\begin{gathered} m+r=5+3=8 \\ 3j=3\times 4=12 \end{gathered}$$

Unstable (since $m+r<3j$)

Summary table

Structure type	Determinacy condition	Stability condition
Beam (2D)	$r=3$	Geometry & supports
Truss (2D)	$m+r=2j$	$m\ge 2j-r$
Frame (2D)	$m+r=3j$	$m+r\ge 3j$

Snow and wind loads

Flat roof snow loads

The flat roof snow load is given by:

$$p_f=0.7C_e{C}_t{I}_s{P}_g$$

Where:

• $C_e$ = exposure factor
• $C_t$ = thermal factor
• $I_s$ = importance factor
• $P_g$ = ground snow load ($lb/f{t}^2$)

Exposure factor, $C_e$

Terrain category	Fully exposed	Partially exposed	Sheltered
B - suburban	0.9	1.0	1.2
C - open terrain	0.9	1.0	1.1
D - open water	0.8	0.9	1.0

• Fully exposed: Roofs exposed on all sides with no shelter afforded by terrain, higher structures, or trees
• Sheltered: Roofs located tightly in among conifers that qualify as obstructions
• Partially exposed: All other cases

Thermal factor, $C_t$

Structure type	$C_t$
All structures except as indicated below	1.0
Unheated and open air structures	1.2
Structures intentionally kept below freezing	1.3

Importance factor, $I_s$

Risk category	Snow, $I_s$	Seismic, $I_e$
I - Low risk	0.8	1.0
II - All others	1.0	1.0
III - Assembly bldgs	1.1	1.25
IV - Essential facilities	1.2	1.5

Wind loads

The velocity pressure at height $z$ is given by:

$$q_z=0.00256K_z{K}_{zt}{K}_d{V}^2({\text{lb/ft}}^2)$$

Where:

• $K_d$ = wind directionality factor = 0.85 (for most structures)
• $K_z$ = velocity pressure exposure coefficient
• $K_{zt}$ = topographic factor (1.0 for flat ground)
• $V$ = basic wind speed ($mph$)

Velocity pressure exposure coefficient, $K_z$

Height above ground (ft)	B - suburban	C - open terrain	D - open water
0-15	0.57	0.85	1.03
20	0.62	0.90	1.08
25	0.66	0.94	1.12

Load combinations for steel member design: ASD vs LRFD

In steel design, ASD (Allowable Strength Design) and LRFD (Load and Resistance Factor Design) use different load levels and different safety formats.

Definitions

ASD checks service-level (unfactored) loads against an allowable strength.

ASD:

$$R_a\le \frac{R_n}{\Omega }$$

• $R_a$: Actual (required) strength due to service loads
• $R_n$: Nominal strength
• $\Omega$: Safety factor (typically $>1.0$)

Definitions

LRFD checks factored (ultimate) loads against a reduced nominal strength.

LRFD:

$$P_u\le \varphi R_n$$

• $P_u$: Factored (ultimate) load
• $R_n$: Nominal strength
• $\varphi$: Resistance factor (typically $<1.0$)

Load combinations

Load combinations define how you compute $R_a$ (ASD) and $P_u$ (LRFD) when multiple load types can act together.

ASD load combinations:

$$\begin{array}{rl}{R}_a& =D+F\\ R_a& =D+H+F+L+T\\ R_a& =D+H+F+(L_r\text{ or }S\text{ or }R)\\ R_a& =D+H+0.75(L+T)+0.75(L_r\text{ or }S\text{ or }R)\\ R_a& =D+H+F+(W\text{ or }0.7E)\\ R_a& =D+H+F+0.75(W\text{ or }0.7E)+0.75L+0.75(L_r\text{ or }S\text{ or }R)\\ R_a& =0.6D+(W\text{ or }0.7E)+H\end{array}$$

LRFD load combinations:

$$\begin{array}{rl}{P}_u& =1.4(D+F)\\ P_u& =1.2(D+F+T)+1.6(L+H)+0.5(L_r\text{ or }S\text{ or }R)\\ P_u& =1.2D+1.6(L_r\text{ or }S\text{ or }R)+(\alpha L\text{ or }0.5W)\\ P_u& =1.2D+1.0W+\alpha L+0.5(L_r\text{ or }S\text{ or }R)\\ P_u& =1.2DL+1.0E+\alpha L+0.2S\\ P_u& =0.9D+(1.0W\text{ or }1.0E)+1.6H\end{array}$$

(When fluid loads $F$, horizontal loads $H$, and self-straining loads $T$ are not present)

ASD (Simplified):

$$\begin{array}{rl}{R}_a& =D\\ R_a& =D+L\\ R_a& =D+(L_r\text{ or }S\text{ or }R)\\ R_a& =D+0.75L+0.75(L_r\text{ or }S\text{ or }R)\\ R_a& =D+(W\text{ or }0.7E)\\ R_a& =D+0.75(W\text{ or }0.7E)+0.75L+0.75(L_r\text{ or }S\text{ or }R)\\ R_a& =0.6D+(W\text{ or }0.7E)\end{array}$$

LRFD (Simplified):

$$\begin{array}{rl}{P}_u& =1.4D\\ P_u& =1.2D+1.6L+0.5(L_r\text{ or }S\text{ or }R)\\ P_u& =1.2D+1.6(L_r\text{ or }S\text{ or }R)+0.5L\text{ or }0.5W\\ P_u& =1.2D+1.0W+0.5L+0.5(L_r\text{ or }S\text{ or }R)\\ P_u& =1.2D+1.0E+0.5L+0.2S\\ P_u& =0.9D+(1.6W\text{ or }1.0E)\end{array}$$

Notation:

Symbol	Load type
$D$	Dead load
$L$	Live load
$F$	Fluid load
$H$	Horizontal earth pressure load
$T$	Self-straining load
$L_r$	Roof live load
$S$	Snow load
$R$	Rain load
$W$	Wind load
$E$	Earthquake load
$\alpha$	Live load factor (commonly 0.5)

Tension member design: ASD vs LRFD

In tension design, you typically check yielding on the gross section and fracture on the effective net section.

ASD:

$$P_a\le \frac{P_n}{{\Omega }_t}$$

LRFD:

$$P_u\le {\varphi }_t{P}_n$$

• $P_a$: Actual service load
• $P_u$: Factored (ultimate) load
• $P_n$: Nominal strength
• ${\Omega }_t$: ASD safety factor
• ${\varphi }_t$: LRFD resistance factor

ASD safety factor ${\Omega }_t$

• Yielding: ${\Omega }_t=1.67$
• Fracture: ${\Omega }_t=2.00$

LRFD resistance factor ${\varphi }_t$

• Yielding: ${\varphi }_t=0.90$
• Fracture: ${\varphi }_t=0.75$

Nominal strength expressions

Yielding limit state:

$$P_n=F_y{A}_g$$

• $F_y$: Yield strength
• $A_g$: Gross area of the member

Fracture limit state:

$$P_n=F_u{A}_e$$

• $F_u$: Ultimate strength
• $A_e=U{A}_n$: Effective net area
• $U$: Shear lag factor
• $A_n$: Net area

Net area calculation

For parallel bolt holes:

$$A_n=\left[b_g-\sum \left(d_h+\frac{1}{16}\right)\right]t$$

For staggered bolt holes:

$$A_n=\left[b_g-\sum \left(d_h+\frac{1}{16}\right)+\sum \frac{s^2}{4g}\right]t$$

• $b_g$: Gross width
• $t$: Thickness
• $d_h$: Nominal hole diameter $=d_b+\frac{1}{16}$
• $s$: Longitudinal spacing between holes
• $g$: Transverse spacing

Effective area $A_e=U{A}_n$

For bolted members:

• Flat bars: $U=1.0$
• Angles: $U=1-\frac{\bar{x}}{L}$

For welded members:

$$U=\{\begin{array}{ll}1.0& \text{(Flat bars/angles with transverse welds)}\\ 1.0& \text{if }L\ge 2w\\ 0.87& \text{if }2w>L\ge 1.5w\\ 0.75& \text{if }1.5w>L>w\\ 1-\frac{\bar{x}}{L}& \text{(Angles with longitudinal welds only)}\end{array}$$

• $w$: Width of flat bar
• $L$: Length of weld
• $\bar{x}$: Distance from centroid to connection

Block shear strength

Resistance factor:

$$\varphi =0.75$$

Shear lag factor:

$$U_{bs}=1.0\text{(flat bars and angles)}$$

Block shear strength:

• $A_{gv}$: Gross area in shear
• $A_{nv}$: Net area in shear
• $A_{nt}$: Net area in tension

$$\varphi T_n=\{\begin{array}{l}0.75F_u\left(0.6A_{nv}+U_{bs}{A}_{nt}\right)\\ \\ 0.75\left(0.6F_y{A}_{gv}+U_{bs}{F}_u{A}_{nt}\right)\end{array}$$

Compression member design: ASD vs LRFD

Compression members are usually controlled by buckling, so the key step is finding the critical stress $F_{cr}$.

ASD:

$$P_a\le \frac{P_n}{{\Omega }_c}$$

LRFD:

$$P_u\le {\varphi }_c{P}_n$$

Where:

• $P_a$: Actual axial service load
• $P_u$: Factored axial load
• $P_n$: Nominal axial compressive strength
• ${\Omega }_c=1.67$: ASD compression safety factor
• ${\varphi }_c=0.9$: LRFD compression resistance factor

Nominal strength expressions

Compressive strength:

$$P_n=F_{cr}{A}_g$$

Where:

• $F_{cr}$: Critical buckling stress
• $A_g$: Gross cross-sectional area

Critical stress: $F_{cr}$

Based on Euler and inelastic buckling criteria:

$$F_{cr}=\{\begin{array}{ll}0.658^{\frac{F_y}{F_e}}{F}_y,& \text{if }\frac{KL}{r}\le 4.71\sqrt{\frac{E}{F_y}}\text{(Inelastic)}\\ 0.877F_e,& \text{if }\frac{KL}{r}>4.71\sqrt{\frac{E}{F_y}}\text{(Elastic)}\end{array}$$

Where:

• $K$: Effective length factor
• $L$: Unsupported length of member
• $r$: Radius of gyration
• $E=29000ksi$: Modulus of elasticity
• $F_y$: Yield strength of the member
• $F_e$: Euler buckling stress

Elastic buckling stress

For elastic buckling (long slender columns), the Euler stress is:

$$F_e=\frac{{\pi }^{2}E}{{\left(\frac{KL}{r}\right)}^2}$$

Flexural member design: ASD vs LRFD

Flexural design compares the required moment to the available flexural strength.

• ASD:

  $$M_a\le \frac{M_n}{{\Omega }_b}$$

• LRFD:

  $$M_u\le {\varphi }_b{M}_n$$

Where:

• $M_a$: Actual service moment
• $M_u$: Factored design moment
• $M_n$: Nominal flexural strength
• ${\Omega }_b=1.67$: ASD bending safety factor
• ${\varphi }_b=0.9$: LRFD bending resistance factor

Nominal strength expressions $M_n$

$$M_n=\{\begin{array}{ll}{M}_p=F_y{Z}_x,& \text{Plastic (compact, full lateral support)}\\ C_b\left[M_p-\left(M_p-0.7F_y{S}_x\right)\left(\frac{L_b-L_p}{L_r-L_p}\right)\right],& \text{Inelastic (lateral-torsional buckling)}\\ F_{cr}{S}_x,& \text{Elastic buckling}\end{array}$$

Where:

• $F_y$: Yield strength
• $Z_x$: Plastic section modulus
• $S_x$: Elastic section modulus
• $L_b$: Laterally unbraced length
• $L_p$: Limit for full plastic bending
• $L_r$: Limit between inelastic and elastic buckling
• $C_b$: Moment gradient factor

Moment gradient factor $C_b$

$$C_b=\frac{12.5M_{max}}{2.5M_{max}+3M_A+4M_B+3M_C}$$

Where $M_A,M_B,M_C$ are moments at quarter points in the unbraced length, and $M_{max}$ is the maximum moment.

Shear member design: ASD vs LRFD

Shear design compares the required shear to the available shear strength of the web.

• ASD:

  $$V_a\le \frac{V_n}{{\Omega }_v}$$

• LRFD:

  $$V_u\le {\varphi }_v{V}_n$$

Where:

• $V_a$: Actual shear force (service)
• $V_u$: Factored shear force
• $V_n$: Nominal shear strength
• ${\Omega }_v=1.67$, ${\varphi }_v=0.9$

Nominal shear strength

$$V_n=0.6F_y{A}_w{C}_v$$

Where:

• $F_y$: Yield strength
• $A_w$: Area of the web
• $C_v$: Shear buckling coefficient