A Beginner's Guide to ASCE 7-10

Chapter 2 - Load Combinations

© 2012, T. Bartlett Quimby

BGASCE7-10 Section 2.2

The Load Combination Equations

Last Revised: 09/27/2016

ASCE 7-10 provides load combination equations for both LRFD and ASD.  The ones that you will use will depend on which of the two design philosophies that have been chosen for your project.

You will note that several of the load combination equations have multiple permeations due to use of "or" or "+" in the equations (both wind, W, and seismic, E, are considered to be + loads).  This is true of both the LRFD and ASD combinations.

Load and Resistance Factor Design

If you chose to use LRFD for your design philosophy, then you are to make sure that your structure is capable of supporting the loads resulting from the seven ASCE 7-05 basic load combination equations. 

LRFD applies load factors to service level loads so that they are safely comparable to member strengths (which are generally inelastic) while maintaining the actual (service) loads in the elastic region.  Member strength (the maximum load that the member will support) is generally between 1.3 to 1.4 times the force that will cause yielding in a member.  These load factors are applied in the load combination equations and vary in magnitude according to the load type. 

The magnitude of the LRFD load factors reflect the predictability of the loads.  For example, the load factor for D is generally lower than the load factor for L in any given equation where there is equal probability of simultaneous occurrence of the full value of each load type.  This is because dead loads are much more predictable than live loads and, hence, do not require as great of a factor of safety.

Example:  Analysis of a structure shows that a particular member supports 5 kips dead load and 6 kips live load.  Using LRFD LC-2, the combined design load equals 1.2 times the dead load plus 1.6 times the live load, or 15.6 kips.  The factor for dead load (1.2) is lower than the factor for live load (1.6) because dead load is more predictable than live load.  The load factors are all greater than 1.0 since we want to compare the result to the ultimate strength of the member instead of the yielding strength of the member yet we don't want yielding to occur.  The ultimate strength is generally about 1.3-1.4 times the yield strength of the member.

Allowable Strength Design

For ASD there are eight basic load combination equations.  You will notice that the large load factors found in the LRFD load combinations are absent from the ASD version of the ASCE 7-05 load combination equations.  Also, the predictability of the loads is not considered.  For example both D and L have the same load factor in equations where they are both likely to occur at full value simultaneously.  The probability associated with accurate load determination is not considered at all in the ASD method.  Hence the major difference between LRFD and ASD.

Example:  Analysis of a structure shows that a particular member supports 5 kips dead load and 6 kips live load.  Using ASD LC-2, the combined design load equals the dead load plus the live load, or 11.0 kips.  The factor for dead load (1.0) is the same as the factor for live load (1.0), hence not accounting for the fact that the dead load is more predictable than the live load.  The result of the load combination equation is then generally compared against the yielding strength of the member to ensure elastic behavior.

The Load Combination Equations

The published load combination equations, modified by the exceptions listed in ASCE 7-10, are:

LRFD

1. 1.4(D+F)
2. 1.2(D+F) + 1.6L + 0.5(L_r or S or R) + (0 or 0.9 or 1.6)^bH
3. 1.2(D+F) + 1.6(L_r or S or R) + ((0.5 or 1.0)^aL or 0.5W) + (0 or 0.9 or 1.6)^bH
4. 1.2(D+F) + (0.5 or 1)^cW + (0 or 1 or 2)^cF_a + (0.5 or 1.0)^aL + 0.5(L_r or S or R) + (0 or 0.9 or 1.6)^bH
5. 1.2(D+F) + E + (0.5 or 1.0)^aL + 0.2S + (0 or 0.9 or 1.6)^bH
6. 0.9D + (0.5 or 1)^cW + (0 or 1 or 2)^cF_a + (0.9 or 1.6)^bH
7. 0.9(D+F) + E + (0 or 0.9 or 1.6)^bH

Foot Notes:

^a Note that the load factor for L in equations (3), (4), and (5) is permitted to equal 0.5 for occupancies in which the unit live load is less than or equal to 100 psf, except for garages or areas occupied as places of public assembly.

^b Note that the load factor for H is 0.9 when it resists the primary load (i.e. has opposite sign) and is permanent.  If H resists the primary load and is not permanent then use a a load factor of 0. The load factor for H is 1.6 when it contributes to the primary load (i.e. has the same sign)

^c The coefficient on W is 0.5 and 1.0 on F_a when the structure is in a noncoastal-A zone with F_a being non-zero.  Otherwise the coefficient on W is 1.0 and 2.0 on F_a.

When atmospheric ice is included, ASCE 7-10 requires modifications to equations (2), (4), and (6), effectively resulting in three new equations which are listed here:

  2_ice.  1.2(D + F) + 1.6L + 0.2D_i + 0.5S + (0 or 0.9 or 1.6)^bH
  4_ice.  1.2D + (0.5 or 1.0)^aL + D_i + W_i + 0.5S
  6_ice.  0.9D +  D_i + W_i + (0 or 0.9 or 1.6)^bH

ASD

1. D + F
2. D + L + (0 or 0.6 or 1.0)^dH + F
3. D + (L_r or S or R) + (0 or 0.6 or 1.0)^dH + F
4. D + 0.75L + 0.75(L_r or S or R) + (0 or 0.6 or 1.0)^dH + F
5. D + (0.6W or (0 or 0.7)^eE) + H + F + (0.75 or 1.5)^eF_a
6a. D + 0.75L + 0.75(0.6W) + 0.75(L_r or S or R) + (0 or 0.6 or 1.0)^dH + F + (0.75 or 1.5)^eF_a
6b. D + 0.75L + 0.75((0 or 0.7)^eE) + 0.75S + (0 or 0.6 or 1.0)^dH + F + (0.75 or 1.5)^eF_a
7. 0.6D + 0.6W + (0 or 0.6 or 1.0)^dH + (0.75 or 1.5)^eF_a
8. 0.6D + 0.7E + (0 or 0.6 or 1.0)^dH + 0.6F

When atmospheric ice is included, ASCE 7-10 requires modifications to equations (2), (3), and (7), effectively resulting in three new equations which are listed here:

2_ice.  D + L + 0.7D_i + (0 or 0.6 or 1.0)^dH + F
3_ice.  D + 0.7D_i + 0.7W_i + S + (0 or 0.6 or 1.0)^dH + F
7_ice.  0.6D + 0.7D_i + 0.7W_i + (0 or 0.6 or 1.0)^dH

Foot Notes:

^d Note that the load factor for H is 0.6 when it resists the primary load (i.e. has opposite sign) and is permanent.  If H resists the primary load and is not permanent then use a load factor of 0. The load factor for H is 1.0 when it contributes to the primary load (i.e. has the same sign)

^e The coefficient on E is 0 in equations 5 and 6b whenever F_a is included. The coefficient on F_a is 0.75 when the structure is in a noncoastal-A zone and F_a is non-zero.  Otherwise the coefficient on F_a is 1.5.

Note that some of the exceptions listed in ASCE 7-10 have been omitted as they are fairly rare or are specialized cases.

For the purposes of this text, we will identify the equations and their permutations by the labels defined as defined in Table 2.1.

Table 2.1
ASCE 7-10 Load Combination Equation Permutations