Building Heights and Building Costs

SAN JOSÉ STATE UNIVERSITY
ECONOMICS DEPARTMENT
Thayer Watkins

Building Heights and Building Costs

The basic problem in the increasing cost of building heights is that a higher building requires a stronger, broader and heavier understructure which adds to the cost of each floor below. There are two problems that require strength:

the compressive strength of the building material and the land substructure upon which it rests
the tensile and compressive strength of the building as a result of the bending induced by wind pressure against the sides of the building
the tensile and compressive strength of the building due to earth quake forces.

For now only the compressive strength of the building will be considered.

Let W_n be the weight of the structure on or above the n-th floor of a building. Likewise let A_n be the area of the n-th floor. Suppose the design criterion is that W_n/A_n has to be equal to k.

W_n = W_n+1 + aA_n where a is the weight per unit area. Note that necessarily a<k because otherwise there would be no way to meet the requirement that the weight per unit area be equal to k.

Since A_n=W_n/k the pertinent relationship is:

(1-a/k)W_n = W_n+1

This means that:

W_n = b^N-nW_N

where b = 1/(1-a/k) and N is the top floor of the building. Since A_n = W_n/a:

A_n = b^N-nA_N

If the cross-section of the building is rectangular then the width and depth of the building are given by:

L_n = [b^1/2]^N-nL_N

where L is any linear cross-sectional dimension. The total area of the building T_N is the sum of the area of the separate floors; i.e.,

T_N = A₁ + A₂ + ... + A_N ==
A_N[b^N-1 + b^N-2 + ... + 1] =
A_N(b^N-1)/(b-1).

The total cost of the building , C_N, is proportional to W₁ and thus proportional to A₁. The relation is:

C_N = cW₁ = caA₁
=caA_Nb^N-1.

The cost per unit area is then:

C_N/T_N = b^N-1(b-1)/[b^N-1]
= (1 - 1/b)/(1 - b^-N)

Let us consider a numerical example.Suppose a=100 lbs/sq.ft and k=400 lbs/sq.ft. This means that b = 1/.75 = 1.333. Consider a ten floor building with the top floor 100 feet wide and 100 feet deep. Given the above values for a and k the dimensions of the building would be:

Floor	Area	Width
1	133183	365
2	99887	316
3	74915	274
4	56187	237
5	42140	205
6	31605	178
7	23704	154
8	17778	133
9	13333	115
10	10,000	100

For a building in which the ratio a/k=10 the profile is as follows:

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(1-a/k)Wn = Wn+1

Wn = bN-nWN

An = bN-nAN

Ln = [b1/2]N-nLN

TN = A1 + A2 + ... + AN == AN[bN-1 + bN-2 + ... + 1] = AN(bN-1)/(b-1).

CN = cW1 = caA1 =caANbN-1.