Section 2.2
The Load Combination Equations
Last Revised:
09/04/2009
ASCE 705 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 705 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 LC2, 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 1.31.4 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 705 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 LC2, 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 are:
LRFD
 1.4(D + F)
 1.2(D + F + T) + 1.6(L + H) + 0.5(L_{r} or S
or R)
 1.2D + 1.6(L_{r} or S or R) + ((0.5 or 1.0)^{*}L or 0.8W)
 1.2D + 1.6W + (0.5 or 1.0)^{*}L + 0.5(L_{r} or S or R)
 1.2D + 1.0E + (0.5 or 1.0)^{*}L + 0.2S
 0.9D + 1.6W + 1.6H
 0.9D + 1.0E + 1.6H
When atmospheric ice is included, ASCE 705 requires modifications to
equations (2), (4), and (6), effectively resulting in three new equations which
are listed here:
2_{ice}. 1.2(D + F + T) + 1.6(L + H) + 0.2D_{i}
+ 0.5S
4_{ice}. 1.2D + (0.5 or 1.0)^{*}L + D_{i} +
W_{i} + 0.5S
6_{ice}. 0.9D + D_{i} + W_{i} + 1.6H
^{*} 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.
ASD
 D + F
 D + H + F + L + T
 D + H + F + (L_{r} or S or R)
 D + H + F + 0.75(L + T) + 0.75(L_{r} or S or
R)
 D + H + F + (W or 0.7E)
 D + H + F + 0.75(W or 0.7E) + 0.75L + 0.75(L_{r}
or S or R)
 0.6D + W + H
 0.6D + 0.7E + H
For the purposes of this text, we will identify the
equations and their permutations by the labels defined as defined in Table 2.1.
When atmospheric ice is included, ASCE 705 requires modifications to
equations (2), (3), and (7), effectively resulting in three new equations which
are listed here:
2_{ice}. D + H + F + L + T + 0.7D_{i}
3_{ice}. D + H + F + 0.7D_{i} + 0.7W_{i} +
S
6_{ice}. 0.6D + 0.7D_{i} + 0.7W_{i} + H
Table 2.1
ASCE 705 Load Combination Equation Permutations
LRFD

ASD 
LRFDLC1 
1.4(D+F) 
LRFDLC2a 
1.2(D + F + T) + 1.6(L + H) + 0.5L_{r} 
LRFDLC2b 
1.2(D + F + T) + 1.6(L + H) + 0.5S 
LRFDLC2c 
1.2(D + F + T) + 1.6(L + H) + 0.5R 
LRFDLC2i 
1.2(D + F + T) + 1.6(L + H) + 0.2D_{i} + 0.5S 
LRFDLC3a 
1.2D + 1.6L_{r} + (0.5 or 1)^{*}L 
LRFDLC3b 
1.2D + 1.6L_{r} + 0.8W 
LRFDLC3c 
1.2D + 1.6S + (0.5 or 1)^{*}L 
LRFDLC3d 
1.2D + 1.6S + 0.8W 
LRFDLC3e 
1.2D + 1.6R + (0.5 or 1)^{*}L 
LRFDLC3f 
1.2D + 1.6R + 0.8W 
LRFDLC4a 
1.2D + 1.6W + (0.5 or 1)^{*}L + .5L_{r} 
LRFDLC4b 
1.2D + 1.6W + (0.5 or 1)^{*}L + .5S 
LRFDLC4c 
1.2D + 1.6W + (0.5 or 1)^{*}L + .5R 
LRFDLC4i 
1.2D + (0.5 or 1.0)*L + D_{i} + W_{i} + 0.5S 
LRFDLC5a 
1.2D + E + (0.5 or 1)^{*}L + 0.2S 
LRFDLC5b 
1.2D  E + (0.5 or 1)^{*}L + 0.2S 
LRFDLC6a 
0.9D + 1.6W + 1.6H 
LRFDLC6b 
0.9D  1.6W + 1.6H 
LRFDLC6i 
0.9D + D_{i} + W_{i} + 1.6H 
LRFDLC7a 
0.9D + E + 1.6H 
LRFDLC7b 
0.9D  E + 1.6H 
* Note that the load factor for L in
LRFD 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. Otherwise the load factor for L equals 1.0.

ASDLC1 
D + F 
ASDLC2 
D + H + F + L + T 
ASDLC2i 
D + H + F + L + T + 0.7D_{i} 
ASDLC3a 
D + H + F + L_{r} 
ASDLC3b 
D + H + F + S 
ASDLC3c 
D + H + F + R 
ASDLC3i 
D + H + F + 0.7D_{i} + 0.7W_{i} + S 
ASDLC4a 
D + H + F + 0.75(L + T) + 0.75L_{r} 
ASDLC4b 
D + H + F + 0.75(L + T) + 0.75S 
ASDLC4c 
D + H + F + 0.75(L + T) + 0.75R 
ASDLC5a 
D + H + F + W 
ASDLC5b 
D + H + F  W 
ASDLC5c 
D + H + F + 0.7E 
ASDLC5d 
D + H + F  0.7E 
ASDLC6a 
D + H + F + 0.75W + 0.75L + 0.75L_{r} 
ASDLC6b 
D + H + F + 0.75W + 0.75L + 0.75S 
ASDLC6c 
D + H + F + 0.75W + 0.75L + 0.75R 
ASDLC6d 
D + H + F  0.75W + 0.75L + 0.75L_{r} 
ASDLC6e 
D + H + F  0.75W + 0.75L + 0.75S 
ASDLC6f 
D + H + F  0.75W + 0.75L + 0.75R 
ASDLC6g 
D + H + F + 0.75(0.7E) + 0.75L + 0.75L_{r} 
ASDLC6h 
D + H + F + 0.75(0.7E) + 0.75L + 0.75S 
ASDLC6i 
D + H + F + 0.75(0.7E) + 0.75L + 0.75R 
ASDLC6j 
D + H + F  0.75(0.7E) + 0.75L + 0.75L_{r} 
ASDLC6k 
D + H + F  0.75(0.7E) + 0.75L + 0.75S 
ASDLC6l 
D + H + F  0.75(0.7E) + 0.75L + 0.75R 
ASDLC6m 
0.6D + 0.7D_{i} + 0.7W_{i} + H 
ASDLC7a 
0.6D + W + H 
ASDLC7b 
0.6D  W + H 
ASDLC8a 
0.6D + 0.7E + H 
ASDLC8b 
0.6D  0.7E + H 

LRFD
