This study investigates router-based multi-agent systems for automating foundation design calculations through intelligent task classification and expert selection. Three approaches were evaluated: single-agent processing, multi-agent designer-checker architecture, and router-based expert selection. Performance assessment utilized baseline models including DeepSeek R1, ChatGPT 4 Turbo, Grok 3, and Gemini 2.5 Pro across shallow foundation and pile design scenarios. The router-based configuration achieved performance scores of 95.00% for shallow foundations and 90.63% for pile design, representing improvements of 8.75 and 3.13 percentage points over standalone Grok 3 performance respectively. The system outperformed conventional agentic workflows by 10.0 to 43.75 percentage points. Grok 3 demonstrated superior standalone performance without external computational tools, indicating advances in direct LLM mathematical reasoning for engineering applications. The dual-tier classification framework successfully distinguished foundation types, enabling appropriate analytical approaches. Results establish router-based multi-agent systems as optimal for foundation design automation while maintaining professional documentation standards. Given safety-critical requirements in civil engineering, continued human oversight remains essential, positioning these systems as advanced computational assistance tools rather than autonomous design replacements in professional practice.